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Quadratic function

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Quartic_function" title="Quartic function">Quartic function</a>.</div>

In <a href="Mathematics" title="Mathematics">mathematics</a>, a quadratic function of a single <a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> of the form<a href="#cite_note-wolfram-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0,}</annotation>

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</math><img src="./16aa6e5c7ca186f44deb74366ea554b878fcc55d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.639ex; height:3.176ex;" alt="{\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0,}" loading="lazy"></dd></dl>

where ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠ is its variable, and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">⁠, and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">⁠ are <a href="Coefficient" title="Coefficient">coefficients</a>. The <a href="Mathematical_expression" class="mw-redirect" title="Mathematical expression">expression</a> ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle ax^{2}+bx+c}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle ax^{2}+bx+c}</annotation>

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</math><img src="./9fd30b57b0f3a144f25302b6292fce83204327bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.629ex; height:2.676ex;" alt="{\displaystyle \textstyle ax^{2}+bx+c}" loading="lazy">⁠, especially when treated as an <a href="Mathematical_object" title="Mathematical object">object</a> in itself rather than as a function, is a quadratic polynomial, a <a href="Polynomial" title="Polynomial">polynomial</a> of degree two. In <a href="Elementary_mathematics" title="Elementary mathematics">elementary mathematics</a> a polynomial and its associated <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial function</a> are rarely distinguished and the terms quadratic function and quadratic polynomial are nearly synonymous and often abbreviated as quadratic.

The <a href="Graph_of_a_function" title="Graph of a function">graph</a> of a <a href="Function_of_a_real_variable" title="Function of a real variable">real</a> single-variable quadratic function is a <a href="Parabola" title="Parabola">parabola</a>. If a quadratic function is <a href="Equation" title="Equation">equated</a> with zero, then the result is a <a href="Quadratic_equation" title="Quadratic equation">quadratic equation</a>. The solutions of a quadratic equation are the <a href="Zero_of_a_function" title="Zero of a function">zeros</a> (or roots) of the corresponding quadratic function, of which there can be two, one, or zero. The solutions are described by the <a href="Quadratic_formula" title="Quadratic formula">quadratic formula</a>.

A quadratic polynomial or quadratic function can involve more than one variable. For example, a two-variable quadratic function of variables ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠ and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">⁠ has the form

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<annotation encoding="application/x-tex">{\displaystyle f(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f,}</annotation>

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</math><img src="./c46feedc673fcbfa9dc1137cb17d206e61709534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.935ex; height:3.176ex;" alt="{\displaystyle f(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f,}" loading="lazy"></dd></dl>

with at least one of ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

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<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

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</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy">⁠, ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}">

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<annotation encoding="application/x-tex">{\displaystyle b}</annotation>

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</math><img src="./f11423fbb2e967f986e36804a8ae4271734917c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}" loading="lazy">⁠, and ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

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</math><img src="./86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">⁠ not equal to zero. In general the zeros of such a quadratic function describe a <a href="Conic_section" title="Conic section">conic section</a> (a <a href="Circle" title="Circle">circle</a> or other <a href="Ellipse" title="Ellipse">ellipse</a>, a <a href="Parabola" title="Parabola">parabola</a>, or a <a href="Hyperbola" title="Hyperbola">hyperbola</a>) in the ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠–⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}">

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<annotation encoding="application/x-tex">{\displaystyle y}</annotation>

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</math><img src="./b8a6208ec717213d4317e666f1ae872e00620a0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" loading="lazy">⁠ plane. A quadratic function can have an arbitrarily large number of variables. The set of its zero form a <a href="Quadric" title="Quadric">quadric</a>, which is a <a href="Surface_(geometry)" class="mw-redirect" title="Surface (geometry)">surface</a> in the case of three variables and a <a href="Hypersurface" title="Hypersurface">hypersurface</a> in general case.

<div class="mw-heading mw-heading2"><h2 id="Etymology">Etymology</h2></div>

The adjective quadratic comes from the <a href="Latin" title="Latin">Latin</a> word <a href="https://en.wiktionary.org/wiki/en:quadratum#Latin" class="extiw external" title="wikt:en:quadratum">quadrātum</a> ("<a href="Square_(geometry)" class="mw-redirect" title="Square (geometry)">square</a>"). A term raised to the second power like ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \textstyle x^{2}}">

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<annotation encoding="application/x-tex">{\displaystyle \textstyle x^{2}}</annotation>

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</math><img src="./848eccd109145ac60de95fe6e94e535a1a812056.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.384ex; height:2.509ex;" alt="{\displaystyle \textstyle x^{2}}" loading="lazy">⁠ is called a <a href="Square_(algebra)" title="Square (algebra)">square</a> in algebra because it is the area of a square with side ⁠<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy">⁠.

<div class="mw-heading mw-heading2"><h2 id="Terminology">Terminology</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Coefficients">Coefficients</h3></div>

The <a href="Coefficients" class="mw-redirect" title="Coefficients">coefficients</a> of a quadratic function are often taken to be <a href="Real_number" title="Real number">real</a> or <a href="Complex_number" title="Complex number">complex numbers</a>, but they may be taken in any <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>, in which case the <a href="Domain_of_a_function" title="Domain of a function">domain</a> and the <a href="Codomain" title="Codomain">codomain</a> are this ring (see <a href="Polynomial_evaluation" title="Polynomial evaluation">polynomial evaluation</a>).

<div class="mw-heading mw-heading3"><h3 id="Degree">Degree</h3></div>

When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". If the degree is less than 2, this may be called a "<a href="Degeneracy_(mathematics)" title="Degeneracy (mathematics)">degenerate case</a>". Usually the context will establish which of the two is meant.

Sometimes the word "order" is used with the meaning of "degree", e.g. a second-order polynomial. However, where the "<a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree of a polynomial</a>" refers to the largest degree of a non-zero term of the polynomial, more typically "order" refers to the lowest degree of a non-zero term of a <a href="Power_series" title="Power series">power series</a>.

<div class="mw-heading mw-heading3"><h3 id="Variables">Variables</h3></div>

A quadratic polynomial may involve a single <a href="Variable_(mathematics)" title="Variable (mathematics)">variable</a> x (the <a href="Univariate" title="Univariate">univariate</a> case), or multiple variables such as x, y, and z (the multivariate case).

<div class="mw-heading mw-heading4"><h4 id="The_one-variable_case">The one-variable case</h4></div>

Any single-variable quadratic polynomial may be written as

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<annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c,}</annotation>

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</math><img src="./8fb8b910705f051903130697415ef9679d69f6a5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.275ex; height:3.009ex;" alt="{\displaystyle ax^{2}+bx+c,}" loading="lazy"></dd></dl>

where x is the variable, and a, b, and c represent the <a href="Coefficient" title="Coefficient">coefficients</a>. Such polynomials often arise in a <a href="Quadratic_equation" title="Quadratic equation">quadratic equation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+bx+c=0.}">

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<annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c=0.}</annotation>

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</math><img src="./70a0e43dfc81e6fea3be4fc96895a8f9ec2966ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.536ex; height:2.843ex;" alt="{\displaystyle ax^{2}+bx+c=0.}" loading="lazy"> The solutions to this equation are called the <a href="Root_of_a_function" class="mw-redirect" title="Root of a function">roots</a> and can be expressed in terms of the coefficients as the <a href="Quadratic_formula" title="Quadratic formula">quadratic formula</a>. Each quadratic polynomial has an associated quadratic function, whose <a href="Graph_of_a_function" title="Graph of a function">graph</a> is a <a href="Parabola" title="Parabola">parabola</a>.

<div class="mw-heading mw-heading4"><h4 id="Bivariate_and_multivariate_cases">Bivariate and multivariate cases</h4></div>

Any quadratic polynomial with two variables may be written as

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<annotation encoding="application/x-tex">{\displaystyle ax^{2}+by^{2}+cxy+dx+ey+f,}</annotation>

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</math><img src="./65bd4881656f89213bac27c6b1abd092807c67a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.23ex; height:3.009ex;" alt="{\displaystyle ax^{2}+by^{2}+cxy+dx+ey+f,}" loading="lazy"></dd></dl>

where x and y are the variables and a, b, c, d, e, f are the coefficients, and one of a, b and c is nonzero. Such polynomials are fundamental to the study of <a href="Conic_section" title="Conic section">conic sections</a>, as the <a href="Implicit_equation" class="mw-redirect" title="Implicit equation">implicit equation</a> of a conic section is obtained by equating to zero a quadratic polynomial, and the <a href="Zero_of_a_function" title="Zero of a function">zeros</a> of a quadratic function form a (possibly degenerate) conic section.

Similarly, quadratic polynomials with three or more variables correspond to <a href="Quadric" title="Quadric">quadric</a> surfaces or <a href="Hypersurface" title="Hypersurface">hypersurfaces</a>.

Quadratic polynomials that have only terms of degree two are called <a href="Quadratic_form" title="Quadratic form">quadratic forms</a>.

<div class="mw-heading mw-heading2"><h2 id="Forms_of_a_univariate_quadratic_function">Forms of a univariate quadratic function</h2></div>

A univariate quadratic function can be expressed in three formats:<a href="#cite_note-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle f(x)=ax^{2}+bx+c}</annotation>

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</math><img src="./66fca4dfe28e7b4a4a336578daaab18c87397073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.145ex; height:3.176ex;" alt="{\displaystyle f(x)=ax^{2}+bx+c}" loading="lazy"> is called the standard form,</li>

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<annotation encoding="application/x-tex">{\displaystyle f(x)=a(x-r_{1})(x-r_{2})}</annotation>

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</math><img src="./9f9933596a211119a8103257a79fe27aa13c4f77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.91ex; height:2.843ex;" alt="{\displaystyle f(x)=a(x-r_{1})(x-r_{2})}" loading="lazy"> is called the factored form, where r1 and r2 are the roots of the quadratic function and the solutions of the corresponding quadratic equation.</li>

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<annotation encoding="application/x-tex">{\displaystyle f(x)=a(x-h)^{2}+k}</annotation>

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</math><img src="./740f52cbddde0949f3d3c04f7b093baa90df661f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.17ex; height:3.176ex;" alt="{\displaystyle f(x)=a(x-h)^{2}+k}" loading="lazy"> is called the vertex form, where h and k are the x and y coordinates of the vertex, respectively.</li></ul>

The coefficient a is the same value in all three forms. To convert the standard form to factored form, one needs only the <a href="Quadratic_formula" title="Quadratic formula">quadratic formula</a> to determine the two roots r1 and r2. To convert the standard form to vertex form, one needs a process called <a href="Completing_the_square" title="Completing the square">completing the square</a>. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.

<div class="mw-heading mw-heading2"><h2 id="Graph_of_the_univariate_function">Graph of the univariate function</h2></div>

Regardless of the format, the graph of a univariate quadratic function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ax^{2}+bx+c}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=ax^{2}+bx+c}</annotation>

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</math><img src="./66fca4dfe28e7b4a4a336578daaab18c87397073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.145ex; height:3.176ex;" alt="{\displaystyle f(x)=ax^{2}+bx+c}" loading="lazy"> is a <a href="Parabola" title="Parabola">parabola</a> (as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=ax^{2}+bx+c}">

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<annotation encoding="application/x-tex">{\displaystyle y=ax^{2}+bx+c}</annotation>

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<ul><li>If a > 0, the parabola opens upwards.</li>

The coefficient a controls the degree of curvature of the graph; a larger magnitude of a gives the graph a more closed (sharply curved) appearance.

The coefficients b and a together control the location of the axis of symmetry of the parabola (also the x-coordinate of the vertex and the h parameter in the vertex form) which is at

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<annotation encoding="application/x-tex">{\displaystyle x=-{\frac {b}{2a}}.}</annotation>

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</math><img src="./3f91b4a21627e90dac5f9f8713df6966ab18efa3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.112ex; height:5.343ex;" alt="{\displaystyle x=-{\frac {b}{2a}}.}" loading="lazy"></dd></dl>

The coefficient c controls the height of the parabola; more specifically, it is the height of the parabola where it intercepts the y-axis.

<div class="mw-heading mw-heading3"><h3 id="Vertex">Vertex</h3></div>

The vertex of a parabola is the place where it turns; hence, it is also called the turning point. If the quadratic function is in vertex form, the vertex is (h, k). Using the method of completing the square, one can turn the standard form

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<annotation encoding="application/x-tex">{\displaystyle f(x)=ax^{2}+bx+c}</annotation>

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into

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x)&=ax^{2}+bx+c\\&=a(x-h)^{2}+k\\&=a\left(x-{\frac {-b}{2a}}\right)^{2}+\left(c-{\frac {b^{2}}{4a}}\right),\\\end{aligned}}}</annotation>

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</math><img src="./812779b1e5f12c60cdd4a48e16100a9615d8fd85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:36.156ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}f(x)&=ax^{2}+bx+c\\&=a(x-h)^{2}+k\\&=a\left(x-{\frac {-b}{2a}}\right)^{2}+\left(c-{\frac {b^{2}}{4a}}\right),\\\end{aligned}}}" loading="lazy"></dd></dl>

so the vertex, (h, k), of the parabola in standard form is

<mrow>

<mrow>

<mfrac>

<mrow>

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</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(-{\frac {b}{2a}},c-{\frac {b^{2}}{4a}}\right).}</annotation>

</semantics>

</math><img src="./3f63a6d37e1080dc1920c314947aa0991596bc5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.601ex; height:6.343ex;" alt="{\displaystyle \left(-{\frac {b}{2a}},c-{\frac {b^{2}}{4a}}\right).}" loading="lazy"><a href="#cite_note-3">[3]</a></dd></dl>

If the quadratic function is in factored form

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=a(x-r_{1})(x-r_{2})}</annotation>

</semantics>

</math><img src="./9f9933596a211119a8103257a79fe27aa13c4f77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.91ex; height:2.843ex;" alt="{\displaystyle f(x)=a(x-r_{1})(x-r_{2})}" loading="lazy"></dd></dl>

the average of the two roots, i.e.,

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {r_{1}+r_{2}}{2}}}</annotation>

</semantics>

</math><img src="./780698d0d3e6165e91a8f3f43246fa495670bfc6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:7.882ex; height:5.009ex;" alt="{\displaystyle {\frac {r_{1}+r_{2}}{2}}}" loading="lazy"></dd></dl>

is the x-coordinate of the vertex, and hence the vertex (h, k) is

<mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<mrow>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left({\frac {r_{1}+r_{2}}{2}},f\left({\frac {r_{1}+r_{2}}{2}}\right)\right).}</annotation>

</semantics>

</math><img src="./625b59bf20005c0d43aa9c63e55bf22939824ff0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.341ex; height:6.176ex;" alt="{\displaystyle \left({\frac {r_{1}+r_{2}}{2}},f\left({\frac {r_{1}+r_{2}}{2}}\right)\right).}" loading="lazy"></dd></dl>

The vertex is also the maximum point if a < 0, or the minimum point if a > 0.

The vertical line

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=h=-{\frac {b}{2a}}}</annotation>

</semantics>

</math><img src="./cb14374df2503029f476a5a529b00da837c6594c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:13.902ex; height:5.343ex;" alt="{\displaystyle x=h=-{\frac {b}{2a}}}" loading="lazy"></dd></dl>

that passes through the vertex is also the axis of symmetry of the parabola.

<div class="mw-heading mw-heading4"><h4 id="Maximum_and_minimum_points">Maximum and minimum points</h4></div>

Using <a href="Calculus" title="Calculus">calculus</a>, the vertex point, being a <a href="Minima_and_maxima" class="mw-redirect" title="Minima and maxima">maximum or minimum</a> of the function, can be obtained by finding the roots of the <a href="Derivative" title="Derivative">derivative</a>:

<msup>

</mrow>

</msup>

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=ax^{2}+bx+c\quad \Rightarrow \quad f'(x)=2ax+b}</annotation>

</semantics>

</math><img src="./1f38745a790d0f4b0ede028d5f2d3a4e086b26f6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.206ex; height:3.176ex;" alt="{\displaystyle f(x)=ax^{2}+bx+c\quad \Rightarrow \quad f'(x)=2ax+b}" loading="lazy"></dd></dl>

x is a root of f '(x) if f '(x) = 0

resulting in

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x=-{\frac {b}{2a}}}</annotation>

</semantics>

</math><img src="./149700f11980672ab7e1d5af4898f0ac67aba29b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.465ex; height:5.343ex;" alt="{\displaystyle x=-{\frac {b}{2a}}}" loading="lazy"></dd></dl>

with the corresponding function value

<msup>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</msup>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=a\left(-{\frac {b}{2a}}\right)^{2}+b\left(-{\frac {b}{2a}}\right)+c=c-{\frac {b^{2}}{4a}},}</annotation>

</semantics>

</math><img src="./ae75782e18c888977da112fdb39478b10e3745f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.609ex; height:6.509ex;" alt="{\displaystyle f(x)=a\left(-{\frac {b}{2a}}\right)^{2}+b\left(-{\frac {b}{2a}}\right)+c=c-{\frac {b^{2}}{4a}},}" loading="lazy"></dd></dl>

so again the vertex point coordinates, (h, k), can be expressed as

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msup>

</mrow>

</msup>

<mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \left(-{\frac {b}{2a}},c-{\frac {b^{2}}{4a}}\right).}</annotation>

</semantics>

</math><img src="./3f63a6d37e1080dc1920c314947aa0991596bc5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.601ex; height:6.343ex;" alt="{\displaystyle \left(-{\frac {b}{2a}},c-{\frac {b^{2}}{4a}}\right).}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Roots_of_the_univariate_function">Roots of the univariate function</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Quadratic_equation" title="Quadratic equation">Quadratic equation</a></div>

<div class="mw-heading mw-heading3"><h3 id="Exact_roots">Exact roots</h3></div>

The <a href="Root_of_a_function" class="mw-redirect" title="Root of a function">roots</a> (or zeros), r1 and r2, of the univariate quadratic function

<mtr>

<mtd>

</mtd>

<mtd>

<msup>

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</msup>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f(x)&=ax^{2}+bx+c\\&=a(x-r_{1})(x-r_{2}),\\\end{aligned}}}</annotation>

</semantics>

</math><img src="./f24721e4f7f5762fb27070fdeec60f6aa3ec7960.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.309ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}f(x)&=ax^{2}+bx+c\\&=a(x-r_{1})(x-r_{2}),\\\end{aligned}}}" loading="lazy"></dd></dl>

are the values of x for which f(x) = 0.

When the <a href="Coefficient" title="Coefficient">coefficients</a> a, b, and c, are <a href="Real_numbers" class="mw-redirect" title="Real numbers">real</a> or <a href="Complex_numbers" class="mw-redirect" title="Complex numbers">complex</a>, the roots are

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{1}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}},}</annotation>

</semantics>

</math><img src="./05a637a668cecbbeb0fb856622c2d8274b940889.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.945ex; height:6.176ex;" alt="{\displaystyle r_{1}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}},}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{2}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}.}</annotation>

</semantics>

</math><img src="./eb5be8119b3a2729fe1685bac6b1fb87486720d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.945ex; height:6.176ex;" alt="{\displaystyle r_{2}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}.}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading3"><h3 id="Upper_bound_on_the_magnitude_of_the_roots">Upper bound on the magnitude of the roots</h3></div>

The <a href="Absolute_value" title="Absolute value">modulus</a> of the roots of a quadratic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax^{2}+bx+c}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ax^{2}+bx+c}</annotation>

</semantics>

</math><img src="./126c6935d3dd9f1c1da0c388ca2799be4f6f237c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:12.629ex; height:2.843ex;" alt="{\displaystyle ax^{2}+bx+c}" loading="lazy"> can be no greater than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\max(|a|,|b|,|c|)}{|a|}}\times \phi ,}">

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {\max(|a|,|b|,|c|)}{|a|}}\times \phi ,}</annotation>

</semantics>

</math><img src="./0fd1b8637843c7075dd0ff71f7ca64c32f8e41d1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.027ex; height:6.509ex;" alt="{\displaystyle {\frac {\max(|a|,|b|,|c|)}{|a|}}\times \phi ,}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi }</annotation>

</semantics>

</math><img src="./72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> is the <a href="Golden_ratio" title="Golden ratio">golden ratio</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1+{\sqrt {5}}}{2}}.}">

<mfrac>

<mrow>

<msqrt>

</msqrt>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1+{\sqrt {5}}}{2}}.}</annotation>

</semantics>

</math><img src="./4be4bdfc614f36203ac94954bede2c870f15617c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.584ex; height:5.843ex;" alt="{\displaystyle {\frac {1+{\sqrt {5}}}{2}}.}" loading="lazy"><a href="#cite_note-5">[5]</a>

<div class="mw-heading mw-heading2"><h2 id="The_square_root_of_a_univariate_quadratic_function">The square root of a univariate quadratic function</h2></div>

The <a href="Square_root" title="Square root">square root</a> of a univariate quadratic function gives rise to one of the four conic sections, <a href="Almost_always" class="mw-redirect" title="Almost always">almost always</a> either to an <a href="Ellipse" title="Ellipse">ellipse</a> or to a <a href="Hyperbola" title="Hyperbola">hyperbola</a>.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a>0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a>0,}</annotation>

</semantics>

</math><img src="./28dc4d9d3d37d7552c7c3b21d641a6559810f25f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.138ex; height:2.509ex;" alt="{\displaystyle a>0,}" loading="lazy"> then the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\pm {\sqrt {ax^{2}+bx+c}}}">

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=\pm {\sqrt {ax^{2}+bx+c}}}</annotation>

</semantics>

</math><img src="./bc7c010f3f18698a781b4597e0fb6b7e5a0e389c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.014ex; height:3.509ex;" alt="{\displaystyle y=\pm {\sqrt {ax^{2}+bx+c}}}" loading="lazy"> describes a hyperbola, as can be seen by squaring both sides. The directions of the axes of the hyperbola are determined by the <a href="Ordinate" class="mw-redirect" title="Ordinate">ordinate</a> of the <a href="Minimum" class="mw-redirect" title="Minimum">minimum</a> point of the corresponding parabola <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{p}=ax^{2}+bx+c.}">

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{p}=ax^{2}+bx+c.}</annotation>

</semantics>

</math><img src="./635466c029133c402bab8e19f475266daa08a017.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.572ex; height:3.343ex;" alt="{\displaystyle y_{p}=ax^{2}+bx+c.}" loading="lazy"> If the ordinate is negative, then the hyperbola's major axis (through its vertices) is horizontal, while if the ordinate is positive then the hyperbola's major axis is vertical.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a<0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a<0,}</annotation>

</semantics>

</math><img src="./da41cabc2fe6c90a53ac67813c30032bd2756e46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.138ex; height:2.509ex;" alt="{\displaystyle a<0,}" loading="lazy"> then the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=\pm {\sqrt {ax^{2}+bx+c}}}">

<msqrt>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=\pm {\sqrt {ax^{2}+bx+c}}}</annotation>

</semantics>

</math><img src="./bc7c010f3f18698a781b4597e0fb6b7e5a0e389c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.014ex; height:3.509ex;" alt="{\displaystyle y=\pm {\sqrt {ax^{2}+bx+c}}}" loading="lazy"> describes either a circle or other ellipse or nothing at all. If the ordinate of the <a href="Maximum" class="mw-redirect" title="Maximum">maximum</a> point of the corresponding parabola

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{p}=ax^{2}+bx+c}</annotation>

</semantics>

</math><img src="./08ca281e6ae31707997ffa15be0dbdb068f417cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.925ex; height:3.343ex;" alt="{\displaystyle y_{p}=ax^{2}+bx+c}" loading="lazy"> is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an <a href="Empty_set" title="Empty set">empty</a> locus of points.

<div class="mw-heading mw-heading2"><h2 id="Iteration">Iteration</h2></div>

To <a href="Iterated_function" title="Iterated function">iterate a function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ax^{2}+bx+c}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=ax^{2}+bx+c}</annotation>

</semantics>

</math><img src="./66fca4dfe28e7b4a4a336578daaab18c87397073.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.145ex; height:3.176ex;" alt="{\displaystyle f(x)=ax^{2}+bx+c}" loading="lazy">, one applies the function repeatedly, using the output from one iteration as the input to the next.

One cannot always deduce the analytic form of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f^{(n)}(x)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{(n)}(x)}</annotation>

</semantics>

</math><img src="./bdc194c5bec806a9ac12374828c715eb14616565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.957ex; height:3.343ex;" alt="{\displaystyle f^{(n)}(x)}" loading="lazy">, which means the nth iteration of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">. (The superscript can be extended to negative numbers, referring to the iteration of the inverse of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> if the inverse exists.) But there are some analytically <a href="Closed-form_expression" title="Closed-form expression">tractable</a> cases.

For example, for the iterative equation

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=a(x-c)^{2}+c}</annotation>

</semantics>

</math><img src="./25303dcd448e4ff6edd0794e9a2ef5529a73a295.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.634ex; height:3.176ex;" alt="{\displaystyle f(x)=a(x-c)^{2}+c}" loading="lazy"></dd></dl>

one has

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=a(x-c)^{2}+c=h^{(-1)}(g(h(x))),}</annotation>

</semantics>

</math><img src="./dc243e3a996169d5d2784e2dec46b195dbc0b1e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.543ex; height:3.343ex;" alt="{\displaystyle f(x)=a(x-c)^{2}+c=h^{(-1)}(g(h(x))),}" loading="lazy"></dd></dl>

where

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)=ax^{2}}</annotation>

</semantics>

</math><img src="./dbc47247c961763acaaffc73456f86be00de4747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.967ex; height:3.176ex;" alt="{\displaystyle g(x)=ax^{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)=x-c.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h(x)=x-c.}</annotation>

</semantics>

</math><img src="./c2c0386d8a9690d938c0d304cfbb70c0ef134ac5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.4ex; height:2.843ex;" alt="{\displaystyle h(x)=x-c.}" loading="lazy"></dd></dl>

So by induction,

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))}</annotation>

</semantics>

</math><img src="./f764872c6ffa71a47dca527175559482b4e7bdf5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.72ex; height:3.343ex;" alt="{\displaystyle f^{(n)}(x)=h^{(-1)}(g^{(n)}(h(x)))}" loading="lazy"></dd></dl>

can be obtained, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g^{(n)}(x)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{(n)}(x)}</annotation>

</semantics>

</math><img src="./80972682034027823f55dea38a2d9ce868983a67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.755ex; height:3.343ex;" alt="{\displaystyle g^{(n)}(x)}" loading="lazy"> can be easily computed as

<msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}.}</annotation>

</semantics>

</math><img src="./18d43610ef5089c6a40bb23fd1376a303e8caecf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.199ex; height:3.343ex;" alt="{\displaystyle g^{(n)}(x)=a^{2^{n}-1}x^{2^{n}}.}" loading="lazy"></dd></dl>

Finally, we have

<msup>

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</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f^{(n)}(x)=a^{2^{n}-1}(x-c)^{2^{n}}+c}</annotation>

</semantics>

</math><img src="./9fef4f15e4bc0eacb8b43df593d73f19ebe801af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.258ex; height:3.343ex;" alt="{\displaystyle f^{(n)}(x)=a^{2^{n}-1}(x-c)^{2^{n}}+c}" loading="lazy"></dd></dl>

as the solution.

See <a href="Topological_conjugacy" title="Topological conjugacy">Topological conjugacy</a> for more detail about the relationship between f and g. And see <a href="Complex_quadratic_polynomial" title="Complex quadratic polynomial">Complex quadratic polynomial</a> for the chaotic behavior in the general iteration.

The <a href="Logistic_map" title="Logistic map">logistic map</a>

<msub>

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</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n+1}=rx_{n}(1-x_{n}),\quad 0\leq x_{0}<1}</annotation>

</semantics>

</math><img src="./4f1625ce848249fc1a32950d130b95b5b1f5a0d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.967ex; height:2.843ex;" alt="{\displaystyle x_{n+1}=rx_{n}(1-x_{n}),\quad 0\leq x_{0}<1}" loading="lazy"></dd></dl>

with parameter 2<r<4 can be solved in certain cases, one of which is <a href="Chaos_(mathematics)" class="mw-redirect" title="Chaos (mathematics)">chaotic</a> and one of which is not. In the chaotic case r=4 the solution is

<msub>

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</msub>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}=\sin ^{2}(2^{n}\theta \pi )}</annotation>

</semantics>

</math><img src="./529a7d6a9b4e03d892d051afc44311874012f121.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.169ex; height:3.176ex;" alt="{\displaystyle x_{n}=\sin ^{2}(2^{n}\theta \pi )}" loading="lazy"></dd></dl>

where the initial condition parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta ={\tfrac {1}{\pi }}\sin ^{-1}(x_{0}^{1/2})}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta ={\tfrac {1}{\pi }}\sin ^{-1}(x_{0}^{1/2})}</annotation>

</semantics>

</math><img src="./dad039cb8fcdc7f048cf2e04dfe53d482d0cc0d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.38ex; height:3.676ex;" alt="{\displaystyle \theta ={\tfrac {1}{\pi }}\sin ^{-1}(x_{0}^{1/2})}" loading="lazy">. For rational <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy">, after a finite number of iterations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation>

</semantics>

</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy"> maps into a periodic sequence. But almost all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> are irrational, and, for irrational <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \theta }</annotation>

</semantics>

</math><img src="./6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation>

</semantics>

</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy"> never repeats itself – it is non-periodic and exhibits <a href="Sensitive_dependence_on_initial_conditions" class="mw-redirect" title="Sensitive dependence on initial conditions">sensitive dependence on initial conditions</a>, so it is said to be chaotic.

The solution of the logistic map when r=2 is

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

<msup>

</mrow>

</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}={\frac {1}{2}}-{\frac {1}{2}}(1-2x_{0})^{2^{n}}}</annotation>

</semantics>

</math><img src="./3bd31011244f6b782f0c4d2db4578bf412fa96ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.862ex; height:5.176ex;" alt="{\displaystyle x_{n}={\frac {1}{2}}-{\frac {1}{2}}(1-2x_{0})^{2^{n}}}" loading="lazy">

for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}\in [0,1)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}\in [0,1)}</annotation>

</semantics>

</math><img src="./0f27e9b5a86221e9f4788c0d7d7d31d7784c17fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.135ex; height:2.843ex;" alt="{\displaystyle x_{0}\in [0,1)}" loading="lazy">. Since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-2x_{0})\in (-1,1)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1-2x_{0})\in (-1,1)}</annotation>

</semantics>

</math><img src="./2175ff33ba10b17fae4bb51a7b3cecd15c9f8a21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.175ex; height:2.843ex;" alt="{\displaystyle (1-2x_{0})\in (-1,1)}" loading="lazy"> for any value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{0}}</annotation>

</semantics>

</math><img src="./86f21d0e31751534cd6584264ecf864a6aa792cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{0}}" loading="lazy"> other than the unstable fixed point 0, the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1-2x_{0})^{2^{n}}}">

<msub>

</mrow>

</msub>

<msup>

<msup>

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</msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1-2x_{0})^{2^{n}}}</annotation>

</semantics>

</math><img src="./08e4e5e41c925914760754acf36b3260dd5f8029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.378ex; height:3.176ex;" alt="{\displaystyle (1-2x_{0})^{2^{n}}}" loading="lazy"> goes to 0 as n goes to infinity, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation>

</semantics>

</math><img src="./7c5ea190699149306d242b70439e663559e3ffbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" loading="lazy"> goes to the stable fixed point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}.}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}.}</annotation>

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</math><img src="./6efe4c7a59c9e443a25a58e03d24544717be5c77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.305ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Bivariate_(two_variable)_quadratic_function">Bivariate (two variable) quadratic function</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Further information: <a href="Quadric" title="Quadric">Quadric</a> and <a href="Quadratic_form" title="Quadratic form">Quadratic form</a></div>

A bivariate quadratic function is a second-degree polynomial of the form

<msup>

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</msup>

<msup>

</mrow>

</msup>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x,y)=Ax^{2}+By^{2}+Cx+Dy+Exy+F,}</annotation>

</semantics>

</math><img src="./a69ee8aff64f7a49c21a14305c1c3ff1c0d5fada.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.837ex; height:3.176ex;" alt="{\displaystyle f(x,y)=Ax^{2}+By^{2}+Cx+Dy+Exy+F,}" loading="lazy"></dd></dl>

where A, B, C, D, and E are fixed <a href="Coefficient" title="Coefficient">coefficients</a> and F is the <a href="Constant_term" title="Constant term">constant term</a>.

Such a function describes a quadratic <a href="Surface_(mathematics)" title="Surface (mathematics)">surface</a>. Setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(x,y)}</annotation>

</semantics>

</math><img src="./29473ed0c4e838ac9dbe074535e507166c0e9101.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.607ex; height:2.843ex;" alt="{\displaystyle f(x,y)}" loading="lazy"> equal to zero describes the intersection of the surface with the plane <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z=0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z=0,}</annotation>

</semantics>

</math><img src="./463359fa7c7563dc29f2079e63195b0035f1ab5a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.996ex; height:2.509ex;" alt="{\displaystyle z=0,}" loading="lazy"> which is a <a href="Locus_(mathematics)" title="Locus (mathematics)">locus</a> of points equivalent to a <a href="Conic_section" title="Conic section">conic section</a>.

<div class="mw-heading mw-heading3"><h3 id="Minimum/maximum">Minimum/maximum</h3></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4AB-E^{2}<0,}">

<msup>

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</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4AB-E^{2}<0,}</annotation>

</semantics>

</math><img src="./a0497fdc0f46982132cef4352fa31f3b898be2ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.266ex; height:3.009ex;" alt="{\displaystyle 4AB-E^{2}<0,}" loading="lazy"> the function has no maximum or minimum; its graph forms a hyperbolic <a href="Paraboloid" title="Paraboloid">paraboloid</a>.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4AB-E^{2}>0,}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4AB-E^{2}>0,}</annotation>

</semantics>

</math><img src="./3af11980d048f01896f8549c99328c47664acc27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.266ex; height:3.009ex;" alt="{\displaystyle 4AB-E^{2}>0,}" loading="lazy"> the function has a minimum if both A > 0 and B > 0, and a maximum if both A < 0 and B < 0; its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{m},y_{m}),}">

<msub>

</mrow>

</msub>

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\displaystyle (x_{m},y_{m}),}</annotation>

</semantics>

</math><img src="./5858d3c4d857f495142cb01530ff131bfa633320.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.309ex; height:2.843ex;" alt="{\displaystyle (x_{m},y_{m}),}" loading="lazy"> where:

<msub>

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</msub>

<mfrac>

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</mrow>

<mrow>

<msup>

</mrow>

</msup>

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</mfrac>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x_{m}=-{\frac {2BC-DE}{4AB-E^{2}}},}</annotation>

</semantics>

</math><img src="./0e0e5337ae44d96425406f2a32ac123549e7b977.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.627ex; height:5.843ex;" alt="{\displaystyle x_{m}=-{\frac {2BC-DE}{4AB-E^{2}}},}" loading="lazy"></dd></dl>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{m}=-{\frac {2AD-CE}{4AB-E^{2}}}.}</annotation>

</semantics>

</math><img src="./d6cf534967b3fc822e7f0dee6d1ace2c803903eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.416ex; height:5.843ex;" alt="{\displaystyle y_{m}=-{\frac {2AD-CE}{4AB-E^{2}}}.}" loading="lazy"></dd></dl>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4AB-E^{2}=0}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4AB-E^{2}=0}</annotation>

</semantics>

</math><img src="./543e6c483c25f24a641dd47668c7c8fc6070bd22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.619ex; height:2.843ex;" alt="{\displaystyle 4AB-E^{2}=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DE-2CB=2AD-CE\neq 0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DE-2CB=2AD-CE\neq 0,}</annotation>

</semantics>

</math><img src="./c5c1885a0a5cf4aea2047135ae2eab77929a3bd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.451ex; height:2.676ex;" alt="{\displaystyle DE-2CB=2AD-CE\neq 0,}" loading="lazy"> the function has no maximum or minimum; its graph forms a parabolic <a href="Cylinder_(geometry)" class="mw-redirect" title="Cylinder (geometry)">cylinder</a>.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4AB-E^{2}=0}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 4AB-E^{2}=0}</annotation>

</semantics>

</math><img src="./543e6c483c25f24a641dd47668c7c8fc6070bd22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.619ex; height:2.843ex;" alt="{\displaystyle 4AB-E^{2}=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle DE-2CB=2AD-CE=0,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle DE-2CB=2AD-CE=0,}</annotation>

</semantics>

</math><img src="./597ea774097b60ec61b5001875bcbdfd35976e27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.451ex; height:2.509ex;" alt="{\displaystyle DE-2CB=2AD-CE=0,}" loading="lazy"> the function achieves the maximum/minimum at a line—a minimum if A>0 and a maximum if A<0; its graph forms a parabolic cylinder.

<ul><li><a href="Quadratic_form" title="Quadratic form">Quadratic form</a></li>

<li><a href="Quadratic_equation" title="Quadratic equation">Quadratic equation</a></li>

<li><a href="Matrix_representation_of_conic_sections" title="Matrix representation of conic sections">Matrix representation of conic sections</a></li>

<li><a href="Quadric" title="Quadric">Quadric</a></li>

<li><a href="Periodic_points_of_complex_quadratic_mappings" title="Periodic points of complex quadratic mappings">Periodic points of complex quadratic mappings</a></li>

<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of mathematical functions</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric Wolfgang. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/QuadraticEquation.html">"Quadratic Equation"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>. Retrieved 2013-01-06.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFHughes_HallettConnallyMcCallum2007" class="citation book cs1"><a href="Deborah_Hughes_Hallett" title="Deborah Hughes Hallett">Hughes Hallett, Deborah J.</a>; Connally, Eric; <a href="William_G._McCallum" title="William G. McCallum">McCallum, William George</a> (2007). College Algebra. <a href="Wiley_(publisher)" title="Wiley (publisher)">John Wiley & Sons Inc.</a> p. 205. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780471271758</bdi>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFColeman1914" class="citation book cs1">Coleman, Percy (1914). Co-ordinate Geometry. Oxford University Press. p. <a rel="nofollow" class="external text" href="https://books.google.com/books?id=TJU5AQAAMAAJ&pg=PA137">137</a>.</cite>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://math.hmc.edu/funfacts/ffiles/10005.1.shtml">"Complex Roots Made Visible – Math Fun Facts"</a>. Retrieved 1 October 2016.</cite>

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <cite id="CITEREFLord2007" class="citation journal cs1">Lord, Nick (2007-11-01). <a rel="nofollow" class="external text" href="https://doi.org/10.2307/40378441">"Golden Bounds for the Roots of Quadratic Equations"</a>. <a href="The_Mathematical_Gazette" title="The Mathematical Gazette">The Mathematical Gazette</a>. 91 (522): 549. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0025557200182324">10.1017/S0025557200182324</a>. <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/40378441">40378441</a>.</cite>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFGlencoe2003" class="citation book cs1">Glencoe, McGraw-Hill (2003). Algebra 1. Glencoe/McGraw Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780078250835</bdi>.</cite></li>

<li><cite id="CITEREFSaxon1991" class="citation book cs1">Saxon, John H. (May 1991). Algebra 2. Saxon Publishers, Incorporated. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780939798629</bdi>.</cite></li></ul>

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</style><div id="Polynomials_and_polynomial_functions_and_polynomial_equations142" style="font-size:114%;margin:0 4em"><a href="Polynomial" title="Polynomial">Polynomials</a> and <a href="Polynomial_function" class="mw-redirect" title="Polynomial function">polynomial functions</a> and <a href="Polynomial_equation" class="mw-redirect" title="Polynomial equation">polynomial equations</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">By <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Zero_polynomial" class="mw-redirect" title="Zero polynomial">Zero polynomial (degree undefined or −1 or −∞)</a></li>

<li><a href="Constant_function" title="Constant function">Constant function (0)</a></li>

<li><a href="Linear_function_(calculus)" title="Linear function (calculus)">Linear function (1)</a>

<ul><li><a href="Linear_equation" title="Linear equation">Linear equation</a></li></ul></li>

<li>

<ul><li><a href="Quadratic_equation" title="Quadratic equation">Quadratic equation</a></li></ul></li>

<li><a href="Cubic_function" title="Cubic function">Cubic function (3)</a>

<ul><li><a href="Cubic_equation" title="Cubic equation">Cubic equation</a></li></ul></li>

<li><a href="Quartic_function" title="Quartic function">Quartic function (4)</a>

<ul><li><a href="Quartic_equation" title="Quartic equation">Quartic equation</a></li></ul></li>

<li><a href="Quintic_function" title="Quintic function">Quintic function (5)</a></li>

<li><a href="Sextic_equation" title="Sextic equation">Sextic equation (6)</a></li>

<li><a href="Septic_equation" title="Septic equation">Septic equation (7)</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">By properties</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Univariate_polynomial" class="mw-redirect" title="Univariate polynomial">Univariate</a></li>

<li><a href="Bivariate_polynomial" class="mw-redirect" title="Bivariate polynomial">Bivariate</a></li>

<li><a href="Multivariate_polynomial" class="mw-redirect" title="Multivariate polynomial">Multivariate</a></li>

<li><a href="Monomial" title="Monomial">Monomial</a></li>

<li><a href="Binomial_(polynomial)" title="Binomial (polynomial)">Binomial</a></li>

<li><a href="Trinomial" title="Trinomial">Trinomial</a></li>

<li><a href="Irreducible_polynomial" title="Irreducible polynomial">Irreducible</a></li>

<li><a href="Square-free_polynomial" title="Square-free polynomial">Square-free</a></li>

<li><a href="Homogeneous_polynomial" title="Homogeneous polynomial">Homogeneous</a></li>

<li><a href="Quasi-homogeneous_polynomial" title="Quasi-homogeneous polynomial">Quasi-homogeneous</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tools and algorithms</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Factorization_of_polynomials" title="Factorization of polynomials">Factorization</a></li>

<li><a href="Polynomial_greatest_common_divisor" title="Polynomial greatest common divisor">Greatest common divisor</a></li>

<li><a href="Polynomial_long_division" title="Polynomial long division">Division</a></li>

<li><a href="Horner's_method" title="Horner's method">Horner's method of evaluation</a></li>

<li><a href="Polynomial_identity_testing" title="Polynomial identity testing">Polynomial identity testing</a></li>

<li><a href="Polynomial_resultant" class="mw-redirect" title="Polynomial resultant">Resultant</a></li>

<li><a href="Discriminant" title="Discriminant">Discriminant</a></li>

<li><a href="Gr%C3%B6bner_basis" title="Gröbner basis">Gröbner basis</a></li></ul>

</div></td></tr></tbody></table></div>

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