Editing Quadratic equation (section)

==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>