Editing Quadratic equation

{{Short description|Polynomial equation of degree two}}

In [[mathematics]], a '''quadratic equation''' ({{etymology|la|{{wikt-lang|la|quadratus}}|[[square (algebra)|square]]}}) is an [[equation]] that can be rearranged in standard form as<ref>{{cite book |title=Intermediate Algebra with Trigonometry |author1=Charles P. McKeague |edition=reprinted |publisher=Academic Press |year=2014 |isbn=978-1-4832-1875-5 |page=219 |url=https://books.google.com/books?id=e4_iBQAAQBAJ}} [https://books.google.com/books?id=e4_iBQAAQBAJ&pg=PA219 Extract of page 219]</ref>
<math display=block>ax^2 + bx + c = 0\,,</math>
where the [[variable (mathematics)|variable]] {{math|''x''}} represents an unknown number, and {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} represent known numbers, where {{math|''a'' ≠ 0}}. (If {{math|''a'' {{=}} 0}} and {{math|''b'' ≠ 0}} then the equation is [[linear equation|linear]], not quadratic.) The numbers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are the ''[[coefficient]]s'' of the equation and may be distinguished by respectively calling them, the ''quadratic coefficient'', the ''linear coefficient'' and the ''constant coefficient'' or ''free term''.<ref>Protters & Morrey: "Calculus and Analytic Geometry. First Course".</ref>

The values of {{mvar|x}} that satisfy the equation are called ''[[solution (mathematics)|solutions]]'' of the equation, and ''[[zero of a function|roots]]'' or ''[[zero of a function|zeros]]'' of the [[quadratic function]] on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a [[double root]]. If all the coefficients are [[real number]]s, there are either two real solutions, or a single real double root, or two [[complex number|complex]] solutions that are [[complex conjugate]]s of each other. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be [[Factorization|factored]] into an equivalent equation<ref>{{cite book |title=Princeton Review SAT Prep, 2021: 5 Practice Tests + Review & Techniques + Online Tools |author1=The Princeton Review |edition= |publisher=Random House Children's Books |year=2020 |isbn=978-0-525-56974-9 |page=360 |url=https://books.google.com/books?id=IrrQDwAAQBAJ}} [https://books.google.com/books?id=IrrQDwAAQBAJ&pg=PA360 Extract of page 360]</ref> 
<math display=block>ax^2+bx+c=a(x-r)(x-s)=0</math>
where {{Mvar|r}} and {{Mvar|s}} are the solutions for {{Mvar|x}}. 

The [[quadratic formula]]
<math display=block>x=\frac{-b\pm\sqrt{b^2-4ac} }{2a}</math>
expresses the solutions in terms of {{mvar|a}}, {{mvar|b}}, and {{mvar|c}}. [[Completing the square]] is one of several ways for deriving the formula.

Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC.<ref>{{cite book |title=Indra's Pearls: The Vision of Felix Klein |author1=David Mumford |author2=Caroline Series |author3=David Wright |edition=illustrated, reprinted |publisher=Cambridge University Press |year=2002 |isbn=978-0-521-35253-6 |page=37 |url=https://books.google.com/books?id=XFE3jmSEfC8C}} [https://books.google.com/books?id=XFE3jmSEfC8C&pg=PA37 Extract of page 37]</ref><ref>{{cite book |title=Mathematics in Action Teachers' Resource Book 4b |author1= |edition=illustrated |publisher=Nelson Thornes |year=1996 |isbn=978-0-17-431439-4 |page=26 |url=https://books.google.com/books?id=HpaBTnefqnkC}} [https://books.google.com/books?id=HpaBTnefqnkC&pg=PA26 Extract of page 26]</ref>

Because the quadratic equation involves only one unknown, it is called "[[univariate]]". The quadratic equation contains only [[exponentiation|powers]] of {{math|''x''}} that are non-negative integers, and therefore it is a [[polynomial equation]]. In particular, it is a [[degree of a polynomial|second-degree]] polynomial equation, since the greatest power is two.

==Solving the quadratic equation==
[[File:Quadratic equation coefficients.png|thumb|right|300px|Figure 1. Plots of quadratic function {{nowrap|''y'' {{=}} ''ax''<sup>2</sup> + ''bx'' + ''c''}}, varying each coefficient separately while the other coefficients are fixed (at values ''a''&nbsp;=&nbsp;1, ''b''&nbsp;=&nbsp;0, ''c''&nbsp;=&nbsp;0)|<!-- Note: The unusual spellings in this alt text (for example, "eh" for the constant "a" ) is intended to aid enunciation by screen readers. Before changing any alt text, please test your changes in multiple screen readers. -->alt=Figure 1. Plots of the quadratic function, y = eh x squared plus b x plus c, varying each coefficient separately while the other coefficients are fixed at values eh = 1, b = 0, c = 0. The left plot illustrates varying c. When c equals 0, the vertex of the parabola representing the quadratic function is centered on the origin, and the parabola rises on both sides of the origin, opening to the top. When c is greater than zero, the parabola does not change in shape, but its vertex is raised above the origin. When c is less than zero, the vertex of the parabola is lowered below the origin. The center plot illustrates varying b. When b is less than zero, the parabola representing the quadratic function is unchanged in shape, but its vertex is shifted to the right of and below the origin. When b is greater than zero, its vertex is shifted to the left of and below the origin. The vertices of the family of curves created by varying b follow along a parabolic curve. The right plot illustrates varying eh. When eh is positive, the quadratic function is a parabola opening to the top. When eh is zero, the quadratic function is a horizontal straight line. When eh is negative, the quadratic function is a parabola opening to the bottom.]]
A quadratic equation whose [[coefficients]] are [[real number]]s can have either zero, one, or two distinct real-valued solutions, also called ''roots''. When there is only one distinct root, it can be interpreted as two roots with the same value, called a [[double root]]. When there are no real roots, the coefficients can be considered as [[complex numbers]] with zero [[imaginary part]], and the quadratic equation still has two complex-valued roots, [[complex conjugate]]s of each-other with a non-zero imaginary part. A quadratic equation whose coefficients are arbitrary complex numbers always has two complex-valued roots which may or may not be distinct.

The solutions of a quadratic equation can be found by several alternative methods.

===Factoring by inspection===
It may be possible to express a quadratic equation {{math|''ax''<sup>2</sup> + ''bx'' + ''c'' {{=}} 0}} as a product {{math|(''px'' + ''q'')(''rx'' + ''s'') {{=}} 0}}. In some cases, it is possible, by simple inspection, to determine values of ''p'', ''q'', ''r,'' and ''s'' that make the two forms equivalent to one another. If the quadratic equation is written in the second form, then the "Zero Factor Property" states that the quadratic equation is satisfied if {{math|''px'' + ''q'' {{=}} 0}} or {{math|''rx'' + ''s'' {{=}} 0}}. Solving these two linear equations provides the roots of the quadratic.

For most students, factoring by inspection is the first method of solving quadratic equations to which they are exposed.<ref name=Washington2000>{{cite book|last=Washington|first=Allyn J.|title=Basic Technical Mathematics with Calculus, Seventh Edition|year=2000|publisher=Addison Wesley Longman, Inc.|isbn=978-0-201-35666-3}}</ref>{{rp|202&ndash;207}} If one is given a quadratic equation in the form {{math|''x''<sup>2</sup> + ''bx'' + ''c'' {{=}} 0}}, the sought factorization has the form {{math|(''x'' + ''q'')(''x'' + ''s'')}}, and one has to find two numbers {{math|''q''}} and {{math|''s''}} that add up to {{math| ''b''}} and whose product is {{math|''c''}} (this is sometimes called "Vieta's rule"<ref>{{citation|title=Numbers|series=Graduate Texts in Mathematics|volume=123|first1=Heinz-Dieter|last1=Ebbinghaus|first2=John H.|last2=Ewing|publisher=Springer|year=1991|isbn=9780387974972|page=77|url=https://books.google.com/books?id=OKcKowxXwKkC&pg=PA77}}.</ref> and is related to [[Vieta's formulas]]). As an example, {{math|''x''<sup>2</sup> + 5''x'' + 6}} factors as {{math|(''x'' + 3)(''x'' + 2)}}. The more general case where {{math|''a''}} does not equal {{math|1}} can require a considerable effort in trial and error guess-and-check, assuming that it can be factored at all by inspection.

Except for special cases such as where {{math|''b'' {{=}} 0}} or {{math|''c'' {{=}} 0}}, factoring by inspection only works for quadratic equations that have rational roots. This means that the great majority of quadratic equations that arise in practical applications cannot be solved by factoring by inspection.<ref name=Washington2000/>{{rp|207}}

===Completing the square===
{{Main|Completing the square}}
[[File:Polynomialdeg2.svg|thumb|right|300px|Figure 2. For the [[quadratic function]] {{math|''y'' {{=}} ''x''<sup>2</sup> &minus; ''x'' &minus; 2}}, the points where the graph crosses the {{math|''x''}}-axis, {{math|''x'' {{=}} −1}} and {{math|''x'' {{=}} 2}}, are the solutions of the quadratic equation {{math|''x''<sup>2</sup> &minus; ''x'' &minus; 2 {{=}} 0}}.
|alt=Figure 2 illustrates an x y plot of the quadratic function f of x equals x squared minus x minus 2. The x-coordinate of the points where the graph intersects the x-axis, x equals &minus;1 and x equals 2, are the solutions of the quadratic equation x squared minus x minus 2 equals zero.]]
The process of completing the square makes use of the algebraic identity
<math display="block">x^2+2hx+h^2 = (x+h)^2,</math>
which represents a well-defined [[algorithm]] that can be used to solve any quadratic equation.<ref name=Washington2000/>{{rp|207}} Starting with a quadratic equation in standard form, {{math|''ax''<sup>2</sup> + ''bx'' + ''c'' {{=}} 0}} 
#Divide each side by {{math|''a''}}, the coefficient of the squared term.
#Subtract the constant term {{math|''c''/''a''}} from both sides.
#Add the square of one-half of {{math|''b''/''a''}}, the coefficient of {{math|''x''}}, to both sides. This "completes the square", converting the left side into a perfect square.
#Write the left side as a square and simplify the right side if necessary.
#Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side.
#Solve each of the two linear equations.

We illustrate use of this algorithm by solving {{math|2''x''<sup>2</sup> + 4''x'' &minus; 4 {{=}} 0}}
<math display="block">2x^2+4x-4=0</math>
<math display="block"> \ x^2+2x-2=0
 </math>
<math display="block">  \ x^2+2x=2</math>
<math display="block"> \ x^2+2x+1=2+1</math>
<math display="block"> \left(x+1 \right)^2=3</math>
<math display="block"> \ x+1=\pm\sqrt{3}</math>
<math display="block"> \ x=-1\pm\sqrt{3}</math>

The [[plus–minus sign|plus–minus symbol "±"]] indicates that both <math display=inline>x=-1+\sqrt{3}</math> and <math display=inline>x=-1-\sqrt{3}</math> are solutions of the quadratic equation.<ref>{{Citation|last=Sterling|first=Mary Jane|title=Algebra I For Dummies|year=2010|publisher=Wiley Publishing|isbn=978-0-470-55964-2|url=https://books.google.com/books?id=2toggaqJMzEC&q=quadratic+formula&pg=PA219|page=219}}</ref>

=== Quadratic formula and its derivation ===
{{Main|Quadratic formula}}
[[Completing the square]] can be used to [[Quadratic formula#Derivations|derive a general formula]] for solving quadratic equations, called the quadratic formula.<ref>{{citation
|title=Schaum's Outline of Theory and Problems of Elementary Algebra
|first1=Barnett
|last1=Rich
|first2=Philip
|last2=Schmidt
|publisher=The McGraw-Hill Companies
|year=2004
|isbn=978-0-07-141083-0
|url=https://books.google.com/books?id=8PRU9cTKprsC}}, [https://books.google.com/books?id=8PRU9cTKprsC&pg=PA291 Chapter 13 §4.4, p. 291]</ref> The [[mathematical proof]] will now be briefly summarized.<ref>Himonas, Alex.  ''[https://books.google.com/books?id=1Mg5u98BnEMC&q=%22left+as+an+exercise%22+and+%22quadratic+formula%22 Calculus for Business and Social Sciences]'', p. 64 (Richard Dennis Publications, 2001).</ref>  It can easily be seen, by [[polynomial expansion]], that the following equation is equivalent to the quadratic equation:
<math display="block">\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}.</math>
Taking the [[square root]] of both sides, and isolating {{math|''x''}}, gives:
<math display="block">x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.</math>

Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as {{math|''ax''<sup>2</sup> + 2''bx'' + ''c'' {{=}} 0}} or {{math|''ax''<sup>2</sup> &minus; 2''bx'' + ''c'' {{=}} 0}}&nbsp;,<ref name="kahan">{{Citation |first=Willian |last=Kahan |title=On the Cost of Floating-Point Computation Without Extra-Precise Arithmetic |url=http://www.cs.berkeley.edu/~wkahan/Qdrtcs.pdf |date=November 20, 2004 |access-date=2012-12-25}}</ref> where {{math|''b''}} has a magnitude one half of the more common one, possibly with opposite sign. These result in slightly different forms for the solution, but are otherwise equivalent.

A number of [[Quadratic formula#Other derivations|alternative derivations]] can be found in the literature.  These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics.

A lesser known quadratic formula, as used in [[Muller's method]], provides the same roots via the equation
<math display="block">x = \frac{2c}{-b \pm \sqrt {b^2-4ac}}.</math> 
This can be deduced from the standard quadratic formula by [[Vieta's formulas]], which assert that the product of the roots is {{math|''c''/''a''}}. It also follows from dividing the quadratic equation by <math>x^2</math> giving <math>cx^{-2}+bx^{-1}+a=0,</math> solving this for <math>x^{-1},</math> and then inverting.

One property of this form is that it yields one valid root when {{math|''a'' {{=}} 0}}, while the other root contains division by zero, because when {{math|''a'' {{=}} 0}}, the quadratic equation becomes a linear equation, which has one root. By contrast, in this case, the more common formula has a division by zero for one root and an [[indeterminate form]] {{math|0/0}} for the other root. On the other hand, when {{math|''c'' {{=}} 0}}, the more common formula yields two correct roots whereas this form yields the zero root and an indeterminate form {{math|0/0}}.

When neither {{mvar|a}} nor {{mvar|c}} is zero, the equality between the standard quadratic formula and Muller's method,
<math display="block">\frac{2c}{-b - \sqrt {b^2-4ac}} = \frac{-b + \sqrt {b^2-4ac}}{2a}\,,</math>
can be verified by [[cross multiplication]], and similarly for the other choice of signs.

===Reduced quadratic equation===
It is sometimes convenient to reduce a quadratic equation so that its [[leading coefficient]] is one. This is done by dividing both sides by {{math|''a''}}, which is always possible since {{math|''a''}} is non-zero.  This produces the ''reduced quadratic equation'':<ref>Alenit͡syn, Aleksandr and Butikov, Evgeniĭ. ''Concise Handbook of Mathematics and Physics'', p. 38 (CRC Press 1997)</ref>

<math display="block">x^2+px+q=0,</math>

where {{math|''p'' {{=}} ''b''/''a''}} and {{math|''q'' {{=}} ''c''/''a''}}. This [[monic polynomial]] equation has the same solutions as the original.

The quadratic formula for the solutions of the reduced quadratic equation, written in terms of its coefficients, is
<math display="block">x = - \frac{p}{2} \pm \sqrt{\left(\frac{p}{2}\right)^2 - q}\,.</math>

===Discriminant===
[[File:Quadratic eq discriminant.svg|thumb|right|Figure 3. Discriminant signs|alt=Figure 3. This figure plots three quadratic functions on a single Cartesian plane graph to illustrate the effects of discriminant values. When the discriminant, delta, is positive, the parabola intersects the {{math|''x''}}-axis at two points. When delta is zero, the vertex of the parabola touches the {{math|''x''}}-axis at a single point. When delta is negative, the parabola does not intersect the {{math|''x''}}-axis at all.]]
In the quadratic formula, the expression underneath the square root sign is called the ''[[discriminant]]'' of the quadratic equation, and is often represented using an upper case {{math|''D''}} or an upper case Greek [[Delta (letter)|delta]]:<ref>'''Δ''' is the initial of the [[Greek language|Greek]] word '''Δ'''ιακρίνουσα, ''Diakrínousa'', discriminant.</ref>
<math display="block">\Delta = b^2 - 4ac.</math>
A quadratic equation with ''real'' coefficients can have either one or two distinct real roots, or two distinct complex roots. In this case the discriminant determines the number and nature of the roots. There are three cases:

*If the discriminant is positive, then there are two distinct roots <math display="block">\frac{-b + \sqrt {\Delta}}{2a} \quad\text{and}\quad \frac{-b - \sqrt {\Delta}}{2a},</math> both of which are real numbers. For quadratic equations with [[rational number|rational]] coefficients, if the discriminant is a [[square number]], then the roots are rational—in other cases they may be [[quadratic irrational]]s.
*If the discriminant is zero, then there is exactly one [[real number|real]] root <math>-\frac{b}{2a},</math> sometimes called a repeated or [[multiple root|double root]] or two equal roots.
*If the discriminant is negative, then there are no real roots. Rather, there are two distinct (non-real) [[complex number|complex]] roots<ref>{{cite book|last1=Achatz|first1=Thomas|last2=Anderson|first2=John G.|last3=McKenzie|first3=Kathleen|title=Technical Shop Mathematics|year=2005|publisher=Industrial Press|isbn=978-0-8311-3086-2|url=https://books.google.com/books?id=YOdtemSmzQQC&q=quadratic+formula&pg=PA276|page=277}}</ref><math display="block">
-\frac{b}{2a} + i \frac{\sqrt {-\Delta}}{2a} \quad\text{and}\quad -\frac{b}{2a} - i \frac{\sqrt {-\Delta}}{2a},
</math> which are [[complex conjugate]]s of each other. In these expressions {{math|''i''}} is the [[imaginary unit]].

Thus the roots are distinct if and only if the discriminant is non-zero, and the roots are real if and only if the discriminant is non-negative.

===Geometric interpretation===
{{quadratic_function_graph_complex_roots.svg}}
The function {{math|''f''(''x'') {{=}} ''ax''<sup>2</sup> + ''bx'' + ''c''}} is a [[quadratic function]].<ref>{{cite book |last=Wharton |first=P. |title=Essentials of Edexcel Gcse Math/Higher |year=2006 |publisher=Lonsdale |isbn=978-1-905-129-78-2|url=https://books.google.com/books?id=LMmKq-feEUoC&q=%22Quadratic+function%22+%22Quadratic+equation%22&pg=PA63 |page=63}}</ref> The graph of any quadratic function has the same general shape, which is called a [[parabola]]. The location and size of the parabola, and how it opens, depend on the values of {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}. If {{math|''a'' &gt; 0}}, the parabola has a minimum point and opens upward. If {{math|''a'' &lt; 0}}, the parabola has a maximum point and opens downward. The extreme point of the parabola, whether minimum or maximum, corresponds to its [[vertex (curve)|vertex]]. The ''{{math|x}}-coordinate'' of the vertex will be located at <math>\scriptstyle x=\tfrac{-b}{2a}</math>, and the ''{{math|y}}-coordinate'' of the vertex may be found by substituting this ''{{math|x}}-value'' into the function. The ''{{math|y}}-intercept'' is located at the point {{math|(0, ''c'')}}.

The solutions of the quadratic equation {{math|''ax''<sup>2</sup> + {{math|''bx''}} + {{math|''c''}} {{=}} 0}} correspond to the [[root of a function|roots]] of the function {{math|''f''(''x'') {{=}} ''ax''<sup>2</sup> + ''bx'' + ''c''}}, since they are the values of {{math|''x''}} for which {{math|''f''(''x'') {{=}} 0}}. If {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are [[real numbers]] and the [[domain of a function|domain]] of {{math|''f''}} is the set of real numbers, then the roots of {{math|''f''}} are exactly the {{math|''x''}}-[[coordinates]] of the points where the graph touches the {{math|''x''}}-axis. If the discriminant is positive, the graph touches the [[x-axis|{{math|''x''}}-axis]] at two points; if zero, the graph touches at one point; and if negative, the graph does not touch the {{math|''x''}}-axis.

===Quadratic factorization===
The term
<math display="block">x - r</math>
is a factor of the polynomial
<math display="block">ax^2+bx+c</math>
if and only if {{math|''r''}} is a [[root of a function|root]] of the quadratic equation
<math display="block">ax^2+bx+c=0.</math>
It follows from the quadratic formula that
<math display="block">ax^2+bx+c = a \left( x - \frac{-b + \sqrt {b^2-4ac}}{2a} \right) \left( x - \frac{-b - \sqrt {b^2-4ac}}{2a} \right).</math>
In the special case {{math|''b''<sup>2</sup> {{=}} 4''ac''}} where the quadratic has only one distinct root (''i.e.'' the discriminant is zero), the quadratic polynomial can be [[Factorization|factored]] as
<math display="block">ax^2+bx+c = a \left( x + \frac{b}{2a} \right)^2.</math>

===Graphical solution===
[[File:Graphical calculation of root of quadratic equation.png|240px|thumb|Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation {{math|2''x''<sup>2</sup> + 4''x'' &minus; 4 {{=}} 0}}. Although the display shows only five significant figures of accuracy, the retrieved value of {{math|''xc''}} is 0.732050807569, accurate to twelve significant figures.]]
[[Image:Visual.complex.root.finding.png|240px|right|thumb|A quadratic function without real root: {{nowrap|''y'' {{=}} (''x'' − 5)<sup>2</sup> + 9}}. The "3" is the imaginary part of the ''x''-intercept. The real part is the ''x''-coordinate of the vertex. Thus the roots are {{nowrap|5 ± 3''i''}}.]]

The solutions of the quadratic equation 
<math display="block">ax^2+bx+c=0</math>
may be deduced from the [[graph of a function|graph]] of the [[quadratic function]]
<math display="block">f(x)=ax^2+bx+c,</math>
which is a [[parabola]].

If the parabola intersects the {{mvar|x}}-axis in two points, there are two real [[zero of a function|roots]], which are the {{mvar|x}}-coordinates of these two points (also called {{mvar|x}}-intercept).

If the parabola is [[tangent]] to the {{mvar|x}}-axis, there is a double root, which is the {{mvar|x}}-coordinate of the contact point between the graph and parabola.

If the parabola does not intersect the {{mvar|x}}-axis, there are two [[complex conjugate]] roots. Although these roots cannot be visualized on the graph, their [[complex number|real and imaginary parts]] can be.<ref name = "Norton1984">{{citation |title=Complex Roots Made Visible |author=Alec Norton, Benjamin Lotto |journal=The College Mathematics Journal |volume=15 |date=June 1984 |pages=248–249 |issue=3 |doi=10.2307/2686333|jstor=2686333 }}</ref>

Let {{mvar|h}} and {{mvar|k}} be respectively the {{mvar|x}}-coordinate and the {{mvar|y}}-coordinate of the vertex of the parabola (that is the point with maximal or minimal {{mvar|y}}-coordinate. The quadratic function may be rewritten
<math display="block"> y = a(x - h)^2 + k.</math>
Let {{mvar|d}} be the distance between the point of {{mvar|y}}-coordinate {{math|2''k''}} on the axis of the parabola, and a point on the parabola with the same {{mvar|y}}-coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). Then the real part of the roots is {{mvar|h}}, and their imaginary part are {{math|±''d''}}. That is, the roots are 
<math display="block">h+id \quad  \text{and} \quad h-id,</math>
or in the case of the example of the figure
<math display="block">5+3i \quad  \text{and} \quad 5-3i.</math>

===Avoiding loss of significance===
Although the quadratic formula provides an exact solution, the result is not exact if [[real number]]s are approximated during the computation, as usual in [[numerical analysis]], where real numbers are approximated by [[floating point number]]s (called "reals" in many [[programming language]]s). In this context, the quadratic formula is not completely [[numerical stability|stable]].

This occurs when the roots have different [[order of magnitude]], or, equivalently, when {{math|''b''<sup>2</sup>}} and  {{math|''b''<sup>2</sup> − 4''ac''}} are close in magnitude. In this case, the subtraction of two nearly equal numbers will cause [[loss of significance]] or [[catastrophic cancellation]] in the smaller root. To avoid this, the root that is smaller in magnitude, {{math|''r''}}, can be computed as <math>(c/a)/R</math> where {{math|''R''}} is the root that is bigger in magnitude. This is equivalent to using the formula

<math display="block">x =\frac{-2c}{b \pm \sqrt {b^2-4ac}}</math>

using the plus sign if <math>b>0</math> and the minus sign if <math>b<0.</math>

A second form of cancellation can occur between the terms {{math|''b''<sup>2</sup>}} and {{math|4''ac''}} of the discriminant, that is when the two roots are very close. This can lead to loss of up to half of correct significant figures in the roots.<ref name="kahan"/><ref name="Higham2002">{{Citation |first=Nicholas |last=Higham |title=Accuracy and Stability of Numerical Algorithms |edition=2nd |publisher=SIAM |year=2002 |isbn=978-0-89871-521-7 |page=10 }}</ref>

==Examples and applications==
[[File:La Jolla Cove cliff diving - 02.jpg|thumb|The trajectory of the cliff jumper is [[parabola|parabolic]] because horizontal displacement is a linear function of time <math>x=v_x t</math>, while vertical displacement is a quadratic function of time <math>y=\tfrac{1}{2} at^2+v_y t+h</math>. As a result, the path follows quadratic equation <math>y=\tfrac{a}{2v_x^2} x^2+\tfrac{v_y}{v_x} x+h</math>, where <math>v_x</math> and <math>v_y</math> are horizontal and vertical components of the original velocity, {{math|a}} is [[Gravity of Earth|gravitational]] [[acceleration]] and {{math|h}} is original height. The {{math|a}} value should be considered negative here, as its direction (downwards) is opposite to the height measurement (upwards).]]
The [[golden ratio]] is found as the positive solution of the quadratic equation <math>x^2-x-1=0.</math>

The equations of the [[circle]] and the other [[conic sections]]&mdash;[[ellipse]]s, [[parabola]]s, and [[hyperbola]]s&mdash;are quadratic equations in two variables.

Given the [[cosine]] or [[sine]] of an angle, finding the cosine or sine of [[Bisection#Angle bisector|the angle that is half as large]] involves solving a quadratic equation.

The process of simplifying expressions involving the [[nested radical|square root of an expression involving the square root of another expression]] involves finding the two solutions of a quadratic equation.

[[Descartes' theorem]] states that for every four kissing (mutually tangent) circles, their [[radius|radii]] satisfy a particular quadratic equation.

The equation given by [[Fuss' theorem]], giving the relation among the radius of a [[bicentric quadrilateral]]'s [[inscribed circle]], the radius of its [[circumscribed circle]], and the distance between the centers of those circles, can be expressed as a quadratic equation for which the distance between the two circles' centers in terms of their radii is one of the solutions. The other solution of the same equation in terms of the relevant radii gives the distance between the circumscribed circle's center and the center of the [[excircle]] of an [[ex-tangential quadrilateral]].

[[Critical point (mathematics)|Critical points]] of a [[cubic function]] and [[inflection point]]s of a [[quartic function]] are found by solving a quadratic equation.

In [[physics]], for [[motion]] with constant [[acceleration]] <math>a</math>, the [[Displacement (geometry)|displacement]] or position <math>x</math> of a moving body can be expressed as a [[quadratic function]] of [[time]] <math>t</math> given the initial position <math>x_0</math> and initial [[velocity]] <math>v_0</math>: <math display="inline">x = x_0 + v_0 t + \frac{1}2 at^2</math>.

In [[chemistry]], the [[pH]] of a [[Solution (chemistry)|solution]] of [[Acid strength#Weak acids|weak acid]] can be calculated from the negative [[Common logarithm|base-10 logarithm]] of the positive root of a quadratic equation in terms of the [[Acid dissociation constant|acidity constant]] and the [[Molar concentration#Formality or analytical concentration|analytical concentration]] of the acid.

==History==
[[Babylonian mathematics|Babylonian mathematicians]], as early as 2000 BC (displayed on [[First Babylonian dynasty|Old Babylonian]] [[clay tablet]]s) could solve problems relating the areas and sides of rectangles. There is evidence dating this algorithm as far back as the [[Third Dynasty of Ur]].<ref name=Friberg2009>{{cite journal|last=Friberg|first=Jöran|title=A Geometric Algorithm with Solutions to Quadratic Equations in a Sumerian Juridical Document from Ur III Umma|journal=Cuneiform Digital Library Journal|year=2009|volume=3|url=http://cdli.ucla.edu/pubs/cdlj/2009/cdlj2009_003.html}}</ref> In modern notation, the problems typically involved solving a pair of simultaneous equations of the form:
<math display="block"> x+y=p,\ \ xy=q, </math>
which is equivalent to the statement that {{mvar|x}} and {{mvar|y}} are the roots of the equation:<ref name=Stillwell2004>{{cite book |last=Stillwell |first=John |title=Mathematics and Its History (2nd ed.) |year=2004 |publisher=Springer |isbn=978-0-387-95336-6}}</ref>{{rp|86}}
<math display="block">z^2+q=pz.</math>

The steps given by Babylonian scribes for solving the above rectangle problem, in terms of {{mvar|x}} and {{mvar|y}}, were as follows:
#Compute half of ''p''.
#Square the result.
#Subtract ''q''.
#Find the (positive) square root using a table of squares.
#Add together the results of steps (1) and (4) to give {{math|''x''}}.
In modern notation this means calculating <math>x = \frac{p}{2} + \sqrt{\left(\frac{p}{2}\right)^2 - q}</math>, which is equivalent to the modern day [[quadratic formula]] for the larger real root (if any) <math>x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}</math> with {{math|1=''a'' = 1}}, {{math|1=''b'' = −''p''}}, and {{math|1=''c'' = ''q''}}.

Geometric methods were used to solve quadratic equations in Babylonia, Egypt, Greece, China, and India. The Egyptian [[Berlin Papyrus 6619|Berlin Papyrus]], dating back to the [[Middle Kingdom of Egypt|Middle Kingdom]] (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation.<ref>{{cite book|title=The Cambridge Ancient History Part 2 Early History of the Middle East|url=https://books.google.com/books?id=slR7SFScEnwC&pg=PA530|year=1971|publisher=Cambridge University Press|isbn=978-0-521-07791-0|page=530}}</ref> Babylonian mathematicians from circa 400 BC and [[Chinese mathematics|Chinese mathematicians]] from circa 200 BC used [[Dissection problem|geometric methods of dissection]] to solve quadratic equations with positive roots.<ref name=Henderson>{{cite web|last=Henderson|first=David W.|title=Geometric Solutions of Quadratic and Cubic Equations |publisher=Mathematics Department, Cornell University |url=http://www.math.cornell.edu/~dwh/papers/geomsolu/geomsolu.html|access-date=28 April 2013}}</ref><ref name=Aitken>{{cite web|last=Aitken|first=Wayne|title=A Chinese Classic: The Nine Chapters|url=http://public.csusm.edu/aitken_html/m330/china/ninechapters.pdf|publisher=Mathematics Department, California State University|access-date=28 April 2013}}</ref> Rules for quadratic equations were given in ''[[The Nine Chapters on the Mathematical Art]]'', a Chinese treatise on mathematics.<ref name=Aitken/><ref>{{cite book|last=Smith|first=David Eugene|title=History of Mathematics|url=https://books.google.com/books?id=uTytJGnTf1kC&pg=PA380|year=1958|publisher=Courier Dover Publications|isbn=978-0-486-20430-7|page=380}}</ref> These early geometric methods do not appear to have had a general formula. [[Euclid]], the [[Greek mathematics|Greek mathematician]], produced a more abstract geometrical method around 300 BC. With a purely geometric approach [[Pythagoras]] and Euclid created a general procedure to find solutions of the quadratic equation. In his work ''[[Arithmetica]]'', the Greek mathematician [[Diophantus]] solved the quadratic equation, but giving only one root, even when both roots were positive.<ref>{{cite book |title=History of Mathematics, Volume 1 |first1=David Eugene |last1=Smith |publisher=Courier Dover Publications |year=1958 |isbn=978-0-486-20429-1 |page=134 |url=https://books.google.com/books?id=12qdOZ0gsWoC}} [https://books.google.com/books?id=12qdOZ0gsWoC&pg=PA134 Extract of page 134]</ref>

In 628 AD, [[Brahmagupta]], an [[Indian mathematics|Indian mathematician]], gave in his book ''[[Brāhmasphuṭasiddhānta]]'' the first explicit (although still not completely general) solution of the quadratic equation {{math|''ax''<sup>2</sup> + ''bx'' {{=}} ''c''}} as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value."<ref>Brāhmasphuṭasiddhānta, Colebrook translation, 1817, page 346; cited by {{cite book |last=Stillwell |first=John |title=Mathematics and Its History (3rd ed.) |series=Undergraduate Texts in Mathematics |year=2010 |publisher=Springer |isbn=978-0-387-95336-6 |page=93 |doi=10.1007/978-1-4419-6053-5}}</ref> This is equivalent to
<math display="block">x = \frac{\sqrt{4ac+b^2}-b}{2a}.</math>
The ''[[Bakhshali Manuscript]]'' written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as linear [[indeterminate equation]]s (originally of type {{math|''ax''/''c'' {{=}} ''y''}}). [[Muhammad ibn Musa al-Khwarizmi]] (9th century) developed a set of formulas that worked for positive solutions. Al-Khwarizmi goes further in providing a full solution to the general quadratic equation, accepting one or two numerical answers for every quadratic equation, while providing geometric [[Mathematical proof|proofs]] in the process.<ref name=Katz2007>{{Cite journal | last1 = Katz | first1 = V. J. | last2 = Barton | first2 = B. | doi = 10.1007/s10649-006-9023-7 | title = Stages in the History of Algebra with Implications for Teaching | journal = Educational Studies in Mathematics | volume = 66 | issue = 2 | pages = 185&ndash;201 | year = 2006 | s2cid = 120363574 }}</ref> He also described the method of completing the square and recognized that the [[discriminant]] must be positive,<ref name=Katz2007/><ref name=Boyer1991/>{{rp|230}} which was proven by his contemporary [['Abd al-Hamīd ibn Turk]] (Central Asia, 9th century) who gave geometric figures to prove that if the discriminant is negative, a quadratic equation has no solution.<ref name=Boyer1991>{{cite book|last=Boyer|first=Carl B.|editor-link=Uta Merzbach|editor-first=Uta C.|editor-last=Merzbach|title=A History of Mathematics|year=1991|publisher=John Wiley & Sons, Inc.|isbn=978-0-471-54397-8|url=https://archive.org/details/historyofmathema00boye}}</ref>{{rp|234}} While al-Khwarizmi himself did not accept negative solutions, later [[Mathematics in medieval Islam|Islamic mathematicians]] that succeeded him accepted negative solutions,<ref name=Katz2007/>{{rp|191}} as well as [[irrational number]]s as solutions.<ref>{{MacTutor|class=HistTopics|id=Arabic_mathematics|title=Arabic mathematics: forgotten brilliance?|year=1999}} "Algebra was a unifying theory which allowed rational numbers, irrational numbers, geometrical magnitudes, etc., to all be treated as "algebraic objects"."</ref> [[Abū Kāmil Shujā ibn Aslam]] (Egypt, 10th century) in particular was the first to accept irrational numbers (often in the form of a [[square root]], [[cube root]] or [[Nth root|fourth root]]) as solutions to quadratic equations or as [[coefficient]]s in an equation.<ref>Jacques Sesiano, "Islamic mathematics", p. 148, in {{citation|title=Mathematics Across Cultures: The History of Non-Western Mathematics|editor1-first=Helaine|editor1-last=Selin|editor1-link=Helaine Selin|editor2-first=Ubiratan|editor2-last=D'Ambrosio|editor2-link=Ubiratan D'Ambrosio|year=2000|publisher=[[Springer Science+Business Media|Springer]]|isbn=978-1-4020-0260-1}}</ref> The 9th century Indian mathematician [[Sridhara]] wrote down rules for solving quadratic equations.<ref>{{cite book|last=Smith|first=David Eugene|title=History of Mathematics|url=https://books.google.com/books?id=12qdOZ0gsWoC&pg=PA280|year=1958|publisher=Courier Dover Publications|isbn=978-0-486-20429-1|page=280}}</ref>

The Jewish mathematician [[Abraham bar Hiyya|Abraham bar Hiyya Ha-Nasi]] (12th century, Spain) authored the first European book to include the full solution to the general quadratic equation.<ref name=Livio2006>{{cite book |last=Livio |first=Mario |title=The Equation that Couldn't Be Solved |year=2006 |publisher=Simon & Schuster |isbn=978-0743258210 |url=https://books.google.com/books?id=veQ9a3nixDUC&q=Abraham+bar+Hiyya+Ha-Nasi+quadratic&pg=PA62}}</ref> His solution was largely based on Al-Khwarizmi's work.<ref name=Katz2007/> The writing of the Chinese mathematician [[Yang Hui]] (1238–1298 AD) is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier [[Liu Yi (mathematician)|Liu Yi]].<ref name=Ron>{{cite book|last=Ronan|first=Colin|title=The Shorter Science and Civilisation in China|url=https://books.google.com/books?id=XsMxmS7NyukC&pg=PA15|year=1985|publisher=Cambridge University Press|isbn=978-0-521-31536-4|page=15}}</ref> By 1545 [[Gerolamo Cardano]] compiled the works related to the quadratic equations. The quadratic formula covering all cases was first obtained by [[Simon Stevin]] in 1594.<ref>{{Citation |title=The Principal Works of Simon Stevin, Mathematics |volume=II-B |first1=D. J. |last1=Struik |first2=Simon |last2=Stevin |publisher=C. V. Swets & Zeitlinger |year=1958 |page=470 |url=http://www.dwc.knaw.nl/pub/bronnen/Simon_Stevin-%5bII_B%5d_The_Principal_Works_of_Simon_Stevin,_Mathematics.pdf}}</ref> In 1637 [[René Descartes]] published ''[[La Géométrie]]'' containing the quadratic formula in the form we know today.

==Advanced topics==

===Alternative methods of root calculation===

====Vieta's formulas====
{{Main|Vieta's formulas}}

''Vieta's formulas'' (named after [[François Viète]]) are the relations
<math display="block"> x_1 + x_2 = -\frac{b}{a}, \quad x_1 x_2 = \frac{c}{a}</math> 
between the roots of a quadratic polynomial and its coefficients. They result from comparing term by term the relation
<math display="block">\left( x - x_1 \right) \left( x-x_2 \right ) = x^2 - \left( x_1+x_2 \right)x +x_1 x_2 = 0</math>
with the equation
<math display="block"> x^2 + \frac ba x +\frac ca = 0.</math>

The first Vieta's formula is useful for graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the [[Quadratic function#Vertex|vertex]],  the vertex's {{math|''x''}}-coordinate is located at the average of the roots (or intercepts). Thus the {{math|''x''}}-coordinate of the vertex is
<math display="block"> x_V = \frac {x_1 + x_2} {2} = -\frac{b}{2a}.</math>
The {{math|''y''}}-coordinate can be obtained by substituting the above result into the given quadratic equation, giving
<math display="block"> y_V = - \frac{b^2}{4a} + c = - \frac{ b^2 - 4ac} {4a}.</math>
Also, these formulas for the vertex can be deduced directly from the formula (see [[Completing the square]])
<math display="block">ax^2+bx+c=a \left(x+\frac b{2a}\right)^2 - \frac{b^2-4ac}{4a}.</math>

For numerical computation, Vieta's formulas provide a useful method for finding the roots of a quadratic equation in the case where one root is much smaller than the other. If {{math|{{!}}''x''<sub>2</sub>{{!}} &lt;&lt; {{!}}''x''<sub>1</sub>{{!}}}}, then {{math|''x''<sub>1</sub> + ''x''<sub>2</sub> ≈ ''x''<sub>1</sub>}}, and we have the estimate:
<math display="block"> x_1 \approx -\frac{b}{a} .</math>
The second Vieta's formula then provides:
<math display="block">x_2 = \frac{c}{a x_1} \approx -\frac{c}{b} .</math>
These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large {{math|''b''}}), which causes [[round-off error]] in a numerical evaluation. The figure shows the difference between{{clarify|reason=without indication on the numerical accuracy, the figure and its discussion are nonsensical. At least the difference with the exact value of the root must also appear.|date=September 2021}} (i)&nbsp;a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii)&nbsp;an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). As the linear coefficient {{math|''b''}} increases, initially the quadratic formula is accurate, and the approximate formula improves in accuracy, leading to a smaller difference between the methods as {{math|''b''}} increases. However, at some point the quadratic formula begins to lose accuracy because of round off error, while the approximate method continues to improve. Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse.

This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see [[Step response]]).

====Trigonometric solution====
In the days before calculators, people would use [[mathematical table]]s—lists of numbers showing the results of calculation with varying arguments—to simplify and speed up computation. Tables of logarithms and trigonometric functions were common in math and science textbooks. Specialized tables were published for applications such as astronomy, celestial navigation and statistics. Methods of numerical approximation existed, called [[prosthaphaeresis]], that offered shortcuts around time-consuming operations such as multiplication and taking powers and roots.<ref name=Ballew2007>{{cite web|last=Ballew|first=Pat|title=Solving Quadratic Equations — By analytic and graphic methods; Including several methods you may never have seen|url=http://www.pballew.net/quadsol.pdf|access-date=18 April 2013|archive-url=https://web.archive.org/web/20110409173024/http://www.pballew.net/quadsol.pdf|archive-date=9 April 2011|url-status=usurped}}</ref> Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in [[celestial mechanics]] calculations.

It is within this context that we may understand the development of means of solving quadratic equations by the aid of [[trigonometric substitution]]. Consider the following alternate form of the quadratic equation,

{{NumBlk||<math display="block">ax^2 + bx \pm c = 0 ,</math>|{{EquationRef|1}}}}

where the sign of the ± symbol is chosen so that {{math|''a''}} and {{math|''c''}} may both be positive. By substituting

{{NumBlk||<math display="block">x = {\textstyle \sqrt{c/a}} \tan\theta </math>|{{EquationRef|2}}}}

and then multiplying through by {{math|cos<sup>2</sup>(''θ'') / ''c''}}, we obtain

{{NumBlk||<math display="block">\sin^2\theta + \frac{b}{\sqrt {ac}} \sin\theta  \cos\theta \pm \cos^2\theta = 0 .</math>|{{EquationRef|3}}}}

Introducing functions of {{math|2''θ''}} and rearranging, we obtain

{{NumBlk||<math display="block"> \tan 2 \theta_n = + 2 \frac{\sqrt{ac}}{b} ,</math>|{{EquationRef|4}}}}

{{NumBlk||<math display="block"> \sin 2 \theta_p = - 2 \frac{\sqrt{ac}}{b} ,</math>|{{EquationRef|5}}}}

where the subscripts {{math|''n''}} and {{math|''p''}} correspond, respectively, to the use of a negative or positive sign in equation {{EquationNote|1|'''[1]'''}}. Substituting the two values of {{math|''θ''<sub>n</sub>}} or {{math|''θ''<sub>p</sub>}} found from equations {{EquationNote|4|'''[4]'''}} or {{EquationNote|5|'''[5]'''}} into {{EquationNote|2|'''[2]'''}} gives the required roots of {{EquationNote|1|'''[1]'''}}. Complex roots occur in the solution based on equation {{EquationNote|5|'''[5]'''}} if the absolute value of {{math|sin 2''θ''<sub>p</sub>}} exceeds unity. The amount of effort involved in solving quadratic equations using this mixed trigonometric and logarithmic table look-up strategy was two-thirds the effort using logarithmic tables alone.<ref name=Seares1945>{{cite journal|last=Seares|first=F. H.|title=Trigonometric Solution of the Quadratic Equation|journal=Publications of the Astronomical Society of the Pacific |year=1945 |volume=57 |issue=339 |page=307&ndash;309 |doi=10.1086/125759 |bibcode=1945PASP...57..307S|doi-access=free }}</ref> Calculating complex roots would require using a different trigonometric form.<ref name=Aude1938>{{cite journal |last=Aude |first=H. T. R. |title=The Solutions of the Quadratic Equation Obtained by the Aid of the Trigonometry |journal=National Mathematics Magazine |year=1938 |volume=13 |issue=3 |pages=118–121 |doi=10.2307/3028750 |jstor=3028750}}</ref>

To illustrate, let us assume we had available seven-place logarithm and trigonometric tables, and wished to solve the following to six-significant-figure accuracy:
<math display="block">4.16130x^2 + 9.15933x - 11.4207 = 0</math>
#A seven-place lookup table might have only 100,000 entries, and computing intermediate results to seven places would generally require interpolation between adjacent entries. 
#<math>\log a = 0.6192290,  \log b = 0.9618637, \log c  = 1.0576927</math>
#<math>2 \sqrt{ac}/b = 2 \times 10^{(0.6192290 + 1.0576927)/2 - 0.9618637} = 1.505314 </math>
#<math>\theta = (\tan^{-1}1.505314) / 2 = 28.20169^{\circ} \text{ or } -61.79831^{\circ} </math>
#<math>\log | \tan \theta | = -0.2706462 \text{ or } 0.2706462</math>
#<math> \log{\textstyle \sqrt{c/a}} = (1.0576927 - 0.6192290) / 2 = 0.2192318</math>
#<math>x_1 = 10^{0.2192318 - 0.2706462} = 0.888353</math> (rounded to six significant figures) <math display="block">x_2 = -10^{0.2192318 + 0.2706462} = -3.08943</math>

====Solution for complex roots in polar coordinates====

If the quadratic equation <math>ax^2+bx+c=0</math> with real coefficients has two complex roots&mdash;the case where <math>b^2-4ac<0,</math> requiring ''a'' and ''c'' to have the same sign as each other&mdash;then the solutions for the roots can be expressed in polar form as<ref>Simons, Stuart, "Alternative approach to complex roots of real quadratic equations", ''Mathematical Gazette'' 93, March 2009, 91–92.</ref>

<math display="block">x_1, \, x_2=r(\cos \theta \pm i\sin \theta), </math>

where <math>r=\sqrt{\tfrac{c}{a}}</math> and <math>\theta =\cos ^{-1}\left(\tfrac{-b}{2\sqrt{ac}}\right).</math>

====Geometric solution====
[[File:LillsQuadratic.svg|thumb|180px|Figure 6. Geometric solution of {{math|''ax''<sup>2</sup> + ''bx'' + ''c'' {{=}} 0}} using Lill's method. Solutions are −AX1/SA, −AX2/SA|alt=Figure 6. Geometric solution of eh x squared plus b x plus c = 0 using Lill's method. The geometric construction is as follows: Draw a trapezoid S Eh B C. Line S Eh of length eh is the vertical left side of the trapezoid. Line Eh B of length b is the horizontal bottom of the trapezoid. Line B C of length c is the vertical right side of the trapezoid. Line C S completes the trapezoid. From the midpoint of line C S, draw a circle passing through points C and S. Depending on the relative lengths of eh, b, and c, the circle may or may not intersect line Eh B. If it does, then the equation has a solution. If we call the intersection points X 1 and X 2, then the two solutions are given by negative Eh X 1 divided by S Eh, and negative Eh X 2 divided by S Eh.]]

The quadratic equation may be solved geometrically in a number of ways. One way is via [[Lill's method]]. The three coefficients {{math|''a''}}, {{math|''b''}}, {{math|''c''}} are drawn with right angles between them as in SA, AB, and BC in Figure&nbsp;6. A circle is drawn with the start and end point SC as a diameter. If this cuts the middle line AB of the three then the equation has a solution, and the solutions are given by negative of the distance along this line from A divided by the first coefficient {{math|''a''}} or SA. If {{math|''a''}} is {{math|1}} the coefficients may be read off directly. Thus the solutions in the diagram are &minus;AX1/SA and &minus;AX2/SA.<ref>{{Citation |title=Graphical Method for finding readily the Real Roots of Numerical Equations of Any Degree |first=William Herbert |last=Bixby |year=1879 |publisher=West Point N. Y.}}</ref>

[[File:CarlyleCircle.svg|thumb|300px|left|Carlyle circle of the quadratic equation ''x''<sup>2</sup>&nbsp;&minus;&nbsp;''sx''&nbsp;+&nbsp;''p''&nbsp;=&nbsp;0.]]
The [[Carlyle circle]], named after [[Thomas Carlyle]], has the property that the solutions of the quadratic equation are the horizontal coordinates of the intersections of the circle with the [[horizontal axis]].<ref name=Wolfram>{{cite web|last=Weisstein|first=Eric W|title=Carlyle Circle|url=http://mathworld.wolfram.com/CarlyleCircle.html|work=From MathWorld—A Wolfram Web Resource|access-date=21 May 2013}}</ref> Carlyle circles have been used to develop [[ruler-and-compass construction]]s of [[regular polygon]]s.

===Generalization of quadratic equation===
The formula and its derivation remain correct if the coefficients {{math|''a''}}, {{math|''b''}} and {{math|''c''}} are [[complex number]]s, or more generally members of any [[field (mathematics)|field]] whose [[characteristic (algebra)|characteristic]] is not {{math|2}}. (In a field of characteristic 2, the element {{math|2''a''}} is zero and it is impossible to divide by it.)

The symbol
<math display="block">\pm \sqrt {b^2-4ac}</math>
in the formula should be understood as "either of the two elements whose square is {{math|''b''<sup>2</sup> &minus; 4''ac''}}, if such elements exist". In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic {{math|2}}. Even if a field does not contain a square root of some number, there is always a quadratic [[extension field]] which does, so the quadratic formula will always make sense as a formula in that extension field.

====Characteristic 2====
In a field of characteristic {{math|2}}, the quadratic formula, which relies on {{math|2}} being a [[unit (ring theory)|unit]], does not hold. Consider the [[monic polynomial|monic]] quadratic polynomial
<math display="block">x^{2} + bx + c</math>
over a field of characteristic {{math|2}}. If {{math|''b'' {{=}} 0}}, then the solution reduces to extracting a square root, so the solution is
<math display="block">x = \sqrt{c}</math>
and there is only one root since
<math display="block">-\sqrt{c} = -\sqrt{c} + 2\sqrt{c} = \sqrt{c}.</math>
In summary,
<math display="block">\displaystyle x^{2} + c = (x + \sqrt{c})^{2}.</math>
See [[quadratic residue]] for more information about extracting square roots in finite fields.

In the case that {{math|''b'' &ne; 0}}, there are two distinct roots, but if the polynomial is [[irreducible polynomial|irreducible]], they cannot be expressed in terms of square roots of numbers in the coefficient field. Instead, define the '''2-root''' {{math|''R''(''c'')}} of {{math|''c''}} to be a root of the polynomial {{math|''x''<sup>2</sup> + ''x'' + ''c''}}, an element of the [[splitting field]] of that polynomial. One verifies that {{math|''R''(''c'') + 1}} is also a root. In terms of the 2-root operation, the two roots of the (non-monic) quadratic {{math|''ax''<sup>2</sup> + ''bx'' + ''c''}} are
<math display="block">\frac{b}{a}R\left(\frac{ac}{b^2}\right)</math>
and
<math display="block">\frac{b}{a}\left(R\left(\frac{ac}{b^2}\right)+1\right).</math>

For example, let {{math|''a''}} denote a multiplicative generator of the group of units of {{math|''F''<sub>4</sub>}}, the [[Galois field]] of order four (thus {{math|''a''}} and {{math|''a'' + 1}} are roots of {{math|''x''<sup>2</sup> + ''x'' + 1}} over {{math|''F''<sub>4</sub>}}. Because {{math|(''a'' + 1)<sup>2</sup> {{=}} ''a''}}, {{math|''a'' + 1}} is the unique solution of the quadratic equation {{math|''x''<sup>2</sup> + ''a'' {{=}} 0}}. On the other hand, the polynomial {{math|''x''<sup>2</sup> + ''ax'' + 1}} is irreducible over {{math|''F''<sub>4</sub>}}, but it splits over {{math|''F''<sub>16</sub>}}, where it has the two roots {{math|''ab''}} and {{math|''ab'' + ''a''}}, where {{math|''b''}} is a root of {{math|''x''<sup>2</sup> + ''x'' + ''a''}} in {{math|''F''<sub>16</sub>}}.

This is a special case of [[Artin–Schreier theory]].

==See also==
* [[Solving quadratic equations with continued fractions]]
* [[Linear equation]]
* [[Cubic function]]
* [[Quartic equation]]
* [[Quintic equation]]
* [[Fundamental theorem of algebra]]

==References==
{{Reflist|30em}}

==External links==
{{Commons category}}
* {{springer|title=Quadratic equation|id=p/q076050}}
* {{MathWorld|title=Quadratic equations|urlname=QuadraticEquation}}
* [http://plus.maths.org/issue29/features/quadratic/index-gifd.html 101 uses of a quadratic equation] {{Webarchive|url=https://web.archive.org/web/20071110232247/http://plus.maths.org/issue29/features/quadratic/index-gifd.html |date=2007-11-10 }}
* [http://plus.maths.org/issue30/features/quadratic/index-gifd.html 101 uses of a quadratic equation: Part II] {{Webarchive|url=https://web.archive.org/web/20071022022143/http://plus.maths.org/issue30/features/quadratic/index-gifd.html |date=2007-10-22 }}

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[[Category:Elementary algebra]]
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