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| {{About|quadratic equations and their solutions|more general information|Quadratic function}}
| | Hi, everybody! <br>I'm French female ;=). <br>I really like Stamp collecting!<br><br>Feel free to surf to my website; [http://Www.Wznord.com/plus/guestbook.php fifa 15 coin generator] |
| {{Infobox physical quantity
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| |image=[[File:Ball Thrown Downward.jpg|200px]]
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| |bgcolour={default}
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| |caption= In this [[physics]] example, a ball with [[uniform acceleration]] {{math|''a''}} ({{math|[[gravitational acceleration|9.8 m/s<sup>2</sup>]]}}) and initial [[velocity]] {{math|''u''}} ({{math|0.196 m/s}}) is seen at intervals of 0.05 second, with distances marked. If the ball moves a distance {{math|''s''}} the time {{math|''t''}} satisfies the equation:
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| :<math>\tfrac{1}{2} at^2+u t-s=0</math>
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| }}
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| In [[elementary algebra]], a '''quadratic equation''' (from the [[Latin]] ''quadratus'' for "[[Square (algebra)|square]]") is any equation having the form
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| :<math>ax^2+bx+c=0</math>
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| where {{math|''x''}} represents an unknown, and {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are [[Constant term|constant]]s with {{math|''a''}} not equal to {{math|0}}. If {{math|''a'' {{=}} 0}}, then the equation is [[linear equation|linear]], not quadratic. The constants {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are called, respectively, the quadratic [[coefficient]], the linear coefficient and the constant or free term.
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| Because the quadratic equation involves only one unknown, it is called "[[univariate]]". The quadratic equation only contains [[exponentiation|powers]] of {{math|''x''}} that are non-negative integers, and therefore it is a [[polynomial equation]], and in particular it is a [[degree of a polynomial|second degree polynomial equation]] since the greatest power is two.
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| Quadratic equations can be solved by a process known in American English as [[Factorization|factoring]] and in other varieties of English as ''factorising'', by [[completing the square]], by using the [[quadratic formula]], or by [[Graph of a function|graphing]]. Solutions to problems equivalent to the quadratic equation were known as early as 2000 BC.
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| ==Solving the quadratic equation==
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| [[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'' = 1, ''b'' = 0, ''c'' = 0)|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. A dotted parabolic line whose vertex is on the origin, and which opens to the bottom, illustrates how 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.]]
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| A quadratic equation with [[real number|real]] or [[complex number|complex]] [[coefficients]] has two solutions, called ''roots''. These two solutions may or may not be distinct, and they may or may not be real.
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| ===Factoring by inspection===
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| 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.
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| 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=0-201-35666-X}}</ref>{{rp|202–207}} If one is given a quadratic equation in the form {{math|''x''<sup>2</sup> + ''px'' + ''q'' {{=}} 0}}, one would seek to find two numbers that add up to {{math|''p''}} and whose product is {{math|''q''}} ("Vieta's Rule"). 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.
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| 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}}
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| ===Completing the square===
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| {{Main|Completing the square}}
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| [[File:Polynomialdeg2.svg|thumb|right|200px|Figure 2. For the [[quadratic function]] {{math|''y'' {{=}} ''x''<sup>2</sup> − ''x'' − 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> − ''x'' − 2 {{=}} 0}}.
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| |alt=Figure 2 illustrates an {{math|''x'' ''y''}} plot of the quadratic function {{math|''f''}} of {{math|''x''}} equals {{math|''x''}} squared minus {{math|''x''}} minus {{math|2}}. The {{math|''x''}}-coordinate of the points where the graph intersects the {{math|''x''}}-axis, {{math|''x'' {{=}} −1}} and {{math|''x'' {{=}} 2}}, are the solutions of the quadratic equation {{math|''x''}} squared minus {{math|''x''}} minus {{math|2}} equals zero.]]
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| The process of completing the square makes use of the algebraic identity
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| :<math>x^2+2xh+h^2 = (x+h)^2,</math>
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| 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}}
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| #Divide each side by {{math|''a''}}, the coefficient of the squared term.
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| #Rearrange the equation so that the constant term {{math|''c''/''a''}} is on the right side.
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| #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.
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| #Write the left side as a square and simplify the right side if necessary.
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| #Produce two linear equations by equating the square root of the left side with the positive and negative square roots of the right side.
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| #Solve the two linear equations.
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| We illustrate use of this algorithm by solving {{math|2''x''<sup>2</sup> + 4''x'' − 4 {{=}} 0}}
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| :<math>1) \ x^2+2x-2=0</math>
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| :<math>2) \ x^2+2x=2</math>
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| :<math>3) \ x^2+2x+1=2+1</math>
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| :<math>4) \ \left(x+1 \right)^2=3</math>
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| :<math>5) \ x+1=\pm\sqrt{3}</math>
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| :<math>6) \ x=-1\pm\sqrt{3}</math>
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| The [[plus-minus sign|plus-minus symbol "±"]] indicates that both {{math|''x'' {{=}} −1 + √3}} and {{math|''x'' {{=}} −1 − √3}} 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=http://books.google.com/?id=2toggaqJMzEC&pg=PA219&dq=quadratic+formula#v=onepage&q=quadratic%20formula&f=false|page=219}}</ref>
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| ===Quadratic formula and its derivation===
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| {{main|Quadratic formula}}
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| [[Completing the square]] can be used to [[Quadratic formula#Derivation of the formula|derive a general formula]] for solving quadratic equations, called the quadratic formula.<ref>{{citation
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| |title=Schaum's Outline of Theory and Problems of Elementary Algebra
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| |first1=Barnett
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| |last1=Rich
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| |first2=Philip
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| |last2=Schmidt
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| |publisher=The McGraw-Hill Companies
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| |year=2004
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| |isbn=0-07-141083-X
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| |url=http://books.google.com/?id=8PRU9cTKprsC}}, [http://books.google.be/books?id=8PRU9cTKprsC&pg=PA291 Chapter 13 §4.4, p. 291]</ref> The [[mathematical proof]] will now be briefly summarized.<ref>Himonas, Alex. ''[http://books.google.com/books?id=1Mg5u98BnEMC&q=%22left+as+an+exercise%22+and+%22quadratic+formula%22&dq=%22left+as+an+exercise%22+and+%22quadratic+formula%22&hl=en&sa=X&ei=6CJbUu2aFMylkQei6YGABA&ved=0CDMQ6AEwATgK 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:
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| :<math>\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}.</math>
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| Taking the [[square root]] of both sides, and isolating {{math|''x''}}, gives:
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| :<math>x=\frac{-b\pm\sqrt{b^2-4ac\ }}{2a}.</math>
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| :Note: Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as {{math|''ax''<sup>2</sup> − 2''bx'' + ''c'' {{=}} 0}} <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 |accessdate=2012-12-25}}</ref> or {{math|''ax''<sup>2</sup> + 2''bx'' + ''c'' {{=}} ''0''}},<ref>{{Citation |url=http://www.proofwiki.org/wiki/Quadratic_Equation#Also_defined_as |title=Quadratic Equation |journal=Proof Wiki |accessdate=2012-12-25}}</ref> where {{math|''b''}} has a magnitude one half of the more common one. These result in slightly different forms for the solution, but are otherwise equivalent.
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| A number of [[Quadratic formula#Other derivations|alternative derivations]] can be found in the literature which either (a) are simpler than the standard completing the square method, (b) represent interesting applications of other frequently used techniques in algebra, or (c) offer insight into other areas of mathematics.
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| ===Reduced quadratic equation===
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| It is sometimes convenient to reduce a quadratic equation to an equation involving two instead of three constant coefficients. This is done by simply dividing both sides by ''a'', which is possible because ''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>
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| :<math>x^2+px+q=0</math>
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| Here ''p'' = ''b''/''a'' and ''q'' = ''c''/''a'' are the only coefficients in the reduced equation, which is also called a [[monic polynomial|monic equation]].
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| It follows from the quadratic formula that the solution to the reduced quadratic equation is
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| :<math>x = -\frac{p}{2} \pm \sqrt{\left(\frac{p}{2}\right)^2 - q}</math>.
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| ===Discriminant===
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| [[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.]]
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| 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>
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| :<math>\Delta = b^2 - 4ac.</math>
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| 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:
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| <ul><li>If the discriminant is positive, then there are two distinct roots
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| :<math>\frac{-b + \sqrt {\Delta}}{2a} \quad\text{and}\quad \frac{-b - \sqrt {\Delta}}{2a},</math>
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| 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.</li>
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| <li>If the discriminant is zero, then there is exactly one [[real number|real]] root
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| :<math>-\frac{b}{2a},</math>
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| sometimes called a repeated or [[multiple root|double root]].</li>
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| <li>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|last=Achatz|first=Thomas|last2=Anderson|first2=John G.|last3=McKenzie|first3=Kathleen|title=Technical Shop Mathematics|year=2005|publisher=Industrial Press|isbn=0-8311-3086-5|url=http://books.google.co.uk/books?id=YOdtemSmzQQC&pg=PA276&dq=quadratic+formula&hl=en&sa=X&ei=mG_8T9-PMuPC0QXOmZigBw&ved=0CEEQ6AEwAQ#v=onepage&q=quadratic%20formula&f=false|page=277}}</ref>
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| :<math> \frac{-b}{2a} + i \frac{\sqrt {-\Delta}}{2a} \quad\text{and}\quad \frac{-b}{2a} - i \frac{\sqrt {-\Delta}}{2a},</math>
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| which are [[complex conjugate]]s of each other. In these expressions {{math|''i''}} is the [[imaginary unit]].</li></ul>
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| 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.
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| ===Geometric interpretation===
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| The function {{math|''f''(''x'') {{=}} ''ax''<sup>2</sup> + ''bx'' + ''c''}} is the [[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=http://books.google.co.uk/books?id=LMmKq-feEUoC&pg=PA63&dq=%22Quadratic+function%22+%22Quadratic+equation%22&hl=en&sa=X&ei=bnT8T-6AKIWX8gP13bCzBw&ved=0CDsQ6AEwAQ#v=onepage&q=%22Quadratic%20function%22%20%22Quadratic%20equation%22&f=false |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, depends on the values of {{math|''a''}}, {{math|''b''}}, and {{math|''c''}}. As shown in Figure 1, if {{math|''a'' > 0}}, the parabola has a minimum point and opens upward. If {{math|''a'' < 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'')}}.
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| 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}}. As shown in Figure 2, if {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} are [[real numbers]] and the [[domain (mathematics)|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. As shown in Figure 3, 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.
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| ===Quadratic factorization===
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| The term
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| :<math>x - r</math>
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| is a factor of the polynomial
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| : <math>ax^2+bx+c</math>
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| if and only if {{math|''r''}} is a [[root of a function|root]] of the quadratic equation
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| : <math>ax^2+bx+c=0.</math>
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| It follows from the quadratic formula that
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| : <math>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>
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| 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
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| :<math>ax^2+bx+c = a \left( x + \frac{b}{2a} \right)^2.</math>
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| ===Graphing for real roots===
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| [[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'' − 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.|alt=Figure 4. Graphing calculator computation of one of the two roots of the quadratic equation {{math|2 ''x''}} squared plus {{math|4 ''x'' minus 4}} equals zero. Although the display shows only five significant figures of accuracy, the retrieved value of {{math|''x''}} is 0.732050807569, accurate to twelve significant figures.]]
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| For most of the 20th century, graphing was rarely mentioned as a method for solving quadratic equations in high school or college algebra texts. Students learned to solve quadratic equations by factoring, completing the square, and applying the quadratic formula. Recently, graphing calculators have become common in schools and graphical methods have started to appear in textbooks, but they are generally not highly emphasized.<ref name=Ballew2007/>
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| Being able to use a graphing calculator to solve a quadratic equation requires the ability to produce a graph of {{math|''y'' {{=}} ''f''(''x'')}}, the ability to scale the graph appropriately to the dimensions of the graphing surface, and the recognition that when {{math|''f''(''x'') {{=}} 0}}, {{math|''x''}} is a solution to the equation. The skills required to solve a quadratic equation on a calculator are in fact applicable to finding the real roots of any arbitrary function.
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| Since an arbitrary function may cross the {{math|''x''}}-axis at multiple points, graphing calculators generally require one to identify the desired root by positioning a cursor at a "guessed" value for the root. (Some graphing calculators require bracketing the root on both sides of the zero.) The calculator then proceeds, by an iterative algorithm, to refine the estimated position of the root to the limit of calculator accuracy.
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| ===Avoiding loss of significance===
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| Although the quadratic formula provides what in principle should be an exact solution, it does not, from a [[numerical analysis]] standpoint, provide a completely stable method for evaluating the roots of a quadratic equation. If the two roots of the quadratic equation vary greatly in absolute magnitude, {{math|''b''}} will be very close in magnitude to <math>\sqrt{b^2-4ac}</math>, and the subtraction of two nearly equal numbers will cause [[loss of significance]] or [[catastrophic cancellation]]. A second form of cancellation can occur between the terms {{math|''b''<sup>2</sup>}} and {{math|−4''ac''}} of the discriminant, which can lead to loss of up to half of correct significant figures.<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>
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| ==History==
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| [[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. In modern notation, the problems typically involved solving a pair of simultaneous equations of the form:
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| :<math> x+y=p,\ \ xy=q </math>
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| which are equivalent to the equation:<ref name=Stillwell2004>{{cite book |last=Stillwell |first=John |title=Mathematics and Its History (2nd ed.) |year=2004 |publisher=Springer |isbn=0-387-95336-1}}</ref>{{rp|86}}
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| :<math>x^2+q=px</math>
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| The steps given by Babylonian scribes for solving the above rectangle problem were as follows:
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| #Compute half of ''p''.
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| #Square the result.
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| #Subtract ''q''.
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| #Find the square root using a table of squares.
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| #Add together the results of steps (1) and (4) to give {{math|''x''}}.
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| Note that step (5) is essentially equivalent to calculating
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| :<math>x = \frac{p}{2} + \sqrt{\left(\frac{p}{2}\right)^2 - q}</math>
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| There is evidence dating this algorithm as far back as the [[Ur III]] dynasty.<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>
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| In the [[Sulba Sutras]] in [[Indian subcontinent|ancient India]] circa 8th century BC quadratic equations of the form {{math|''ax''<sup>2</sup> {{=}} ''c''}} and {{math|''ax''<sup>2</sup> + ''bx'' {{=}} ''c''}} were explored using geometric methods. 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, but do not appear to have had a general formula.<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|accessdate=28 April 2013}}</ref><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|accessdate=28 April 2013}}</ref>
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| [[Euclid]], the [[Greek mathematics|Greek mathematician]], produced a more abstract geometrical method around 300 BC. [[Pythagoras]] and Euclid used a strictly geometric approach, and found a general procedure to solve 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>David Eugene Smith (1958). "''[http://books.google.com/books?id=12qdOZ0gsWoC&pg=PA134&dq&hl=en#v=onepage&q=&f=false History of mathematics]''". Courier Dover Publications. p.134. ISBN 0-486-20429-4</ref>
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| In 628 AD, [[Brahmagupta]], an [[Indian mathematics|Indian mathematician]], gave the first explicit (although still not completely general) solution of the quadratic equation {{math|''ax''<sup>2</sup> + ''bx'' {{=}} ''c''}} as follows:
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| :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. (''Brahmasphutasiddhanta'', Colebrook translation, 1817, page 346)<ref name=Stillwell2004/>{{rp|87}}
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| This is equivalent to:
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| :<math>x = \frac{\sqrt{4ac+b^2}-b}{2a}.</math>
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| The ''[[Bakhshali Manuscript]]'' written in India in the 7th century AD contained an algebraic formula for solving quadratic equations, as well as quadratic [[indeterminate (variable)|indeterminate]] equations (originally of type {{math|''ax''/''c'' {{=}} ''y''}}
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| [[Muhammad ibn Musa al-Khwarizmi]] ([[Persia]], 9th century), inspired by Brahmagupta, 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 doi|10.1007/s10649-006-9023-7}}</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.; Uta C. Merzbach, rev. editor|title=A History of Mathematics|year=1991|publisher=John Wiley & Sons, Inc.|isbn=0-471-54397-7}}</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=1-4020-0260-2}}</ref>
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| The Indian mathematician [[Sridhara]], who flourished in the 9th and 10th centuries AC provided the modern solution of the quadratic equation.
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| 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=0743258215 |url=http://books.google.com/books?id=veQ9a3nixDUC&pg=PA62&lpg=PA62&dq=Abraham+bar+Hiyya+Ha-Nasi+quadratic&source=bl&ots=85JwJi8y4q&sig=UvI5MOdfntTgYwJgR_-5yEZuvEI&hl=en&ei=yGSjSe7eB4iQngf9p52kBQ&sa=X&oi=book_result&resnum=2&ct=result}}</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]].
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| 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. The first appearance of the general solution in the modern mathematical literature appeared in an 1896 paper by Henry Heaton.<ref name="heaton-1896">Heaton, H. (1896) ''[http://www.jstor.org/stable/info/2971099 A Method of Solving Quadratic Equations]'', [[American Mathematical Monthly]] '''3'''(10), 236–237.</ref>
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| ==Advanced topics==
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| ===Alternative methods of root calculation===
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| ====Vieta's formulas====
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| {{Main|Vieta's formulas}}
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| [[File:Excel quadratic error.PNG|thumb |350px|Figure 5. Graph of the difference between Vieta's approximation for the smaller of the two roots of the quadratic equation {{math|''x''<sup>2</sup> + ''bx'' + ''c'' {{=}} 0}} compared with the value calculated using the quadratic formula. Vieta's approximation is inaccurate for small {{math|''b''}} but is accurate for large {{math|''b''}}. The direct evaluation using the quadratic formula is accurate for small {{math|''b''}} with roots of comparable value but experiences loss of significance errors for large {{math|''b''}} and widely spaced roots. The difference between Vieta's approximation ''versus'' the direct computation reaches a minimum at the large dots, and rounding causes squiggles in the curves beyond this minimum.|alt=Figure 5. Graph of the difference between Vieta's approximation for the smaller of the two roots of the quadratic equation x squared plus b x plus c equals zero compared with the value calculated using the quadratic formula. The difference is plotted as a function of b for two different values of c, c equals 4, and c equals 400,000. The graph is a log log graph, with the vertical axis, the difference, ranging from ten to the minus 13 at the bottom to ten to the minus 1 at the top. The horizontal axis, b, ranges from 10 at the left to ten to the eighth at the right. Vieta's approximation for the smaller root is not accurate for small b but is accurate for large b. The direct evaluation of the smaller root using the quadratic formula is accurate for small b with roots of comparable value, but experiences loss of significance errors for large b and widely spaced roots. When c equals 4, Vieta's approximation starts off poorly at the left, but gets better with larger b, the difference between Vieta's approximation and the quadratic formula reaching a minimum at approximately b equals ten to the fifth. Vieta's approximation and the quadratic formula then start diverging again because the quadratic formula experiences loss of significance error. When c equals four hundred thousand, the difference between Vieta's approximation and the quadratic formula reaches a minimum at approximately b equals ten to the seventh. The curves are both straight to the left of the minimum, indicating a simple monomial power relationship between the difference and b. Likewise, the curves are both approximately straight to the right of the minimum, indicating a power relationship, except that the straight lines have squiggles in them due to the loss of significance errors in the quadratic formula.]]
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| Vieta's formulas give a simple relation between the roots of a polynomial and its coefficients. In the case of the quadratic polynomial, they take the following form:
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| :<math> x_1 + x_2 = -\frac{b}{a} </math>
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| and
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| :<math> x_1 \ x_2 = \frac{c}{a}.</math>
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| These results follow immediately from the relation:
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| :<math>\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>
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| which can be compared term by term with
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| :<math> x^2 + (b/a)x +c/a = 0.</math>
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| The first formula above yields a convenient expression when graphing a quadratic function. Since the graph is symmetric with respect to a vertical line through the [[Quadratic function#Vertex|vertex]], when there are two real roots 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 given by the expression
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| :<math> x_V = \frac {x_1 + x_2} {2} = -\frac{b}{2a}.</math>
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| The {{math|''y''}}-coordinate can be obtained by substituting the above result into the given quadratic equation, giving
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| :<math> y_V = - \frac{b^2}{4a} + c = - \frac{ b^2 - 4ac} {4a}.</math>
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| As a practical matter, Vieta's formulas provide a useful method for finding the roots of a quadratic in the case where one root is much smaller than the other. If {{math|{{!}} ''x'' <sub>2</sub>{{!}} << {{!}} ''x'' <sub>1</sub>{{!}}}}, then {{math|''x'' <sub>1</sub> + ''x'' <sub>2</sub> ≈ ''x'' <sub>1</sub>}}, and we have the estimate:
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| :<math> x_1 \approx -\frac{b}{a} .</math>
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| The second Vieta's formula then provides:
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| :<math>x_2 = \frac{c}{a \ x_1} \approx -\frac{c}{b} .</math>
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| 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. Figure 5 shows the difference between (i) a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii) 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.
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| This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see [[step response]]).
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| ====Trigonometric solution====
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| 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|accessdate=18 April 2013}}</ref> Astronomers, especially, were concerned with methods that could speed up the long series of computations involved in [[celestial mechanics]] calculations.
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| 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,
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| '''[1]''' <math>ax^2 + bx \pm c = 0 ,</math>
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| where the sign of the ± symbol is chosen so that {{math|''a''}} and {{math|''c''}} may both be positive. By substituting
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| '''[2]''' <math>x = \sqrt{c/a} \tan\theta </math>
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| and then multiplying through by {{math|cos<sup>2</sup>''θ''}}, we obtain
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| '''[3]''' <math>\sin^2\theta + \frac{b}{\sqrt {ac}} \sin\theta \cos\theta \pm \cos^2\theta = 0 .</math>
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| Introducing functions of {{math|2''θ''}} and rearranging, we obtain
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| '''[4]''' <math> \tan 2 \theta_n = + 2 \frac{\sqrt{ac}}{b} ,</math>
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| '''[5]''' <math> \sin 2 \theta_p = - 2 \frac{\sqrt{ac}}{b} ,</math>
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| where the subscripts {{math|''n''}} and {{math|''p''}} correspond, respectively, to the use of a negative or positive sign in equation '''[1]'''. Substituting the two values of {{math|''θ''<sub>n</sub>}} or {{math|''θ''<sub>p</sub>}} found from equations '''[4]''' or '''[5]''' into '''[2]''' gives the required roots of '''[1]'''. Complex roots occur in the solution based on equation '''[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=307–309 |url=http://adsabs.harvard.edu/full/1945PASP...57..307S |accessdate=18 April 2013}}</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 |page=118–121 |url=http://www.jstor.org/stable/3028750 |accessdate=20 April 2013}}</ref>
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| :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:
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| :::<math>4.16130x^2 + 9.15933x - 11.4207 = 0</math>
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| #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.
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| #<math>\log a = 0.6192290, \log b = 0.9618637, \log c = 1.0576927</math>
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| #<math>2 \sqrt{ac}/b = 2 \times 10^{(0.6192290 + 1.0576927)/2 - 0.9618637} = 1.505314 </math>
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| #<math>\theta = (\tan^{-1}1.505314) / 2 = 28.20169^{\circ} \text{ or } -61.79831^{\circ} </math>
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| #<math>\log \left\vert \tan \theta \right\vert = -0.2706462 \text{ or } 0.2706462</math>
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| #<math> \log\sqrt{c/a} = (1.0576927 - 0.6192290) / 2 = 0.2192318</math>
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| #<math>x_1 = 10^{0.2192318 - 0.2706462} = 0.888353</math> (rounded to six significant figures)
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| ::<math>x_2 = -10^{0.2192318 + 0.2706462} = -3.08943</math>
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| ====Geometric solution====
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| [[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.]]
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| 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 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 −AX1/SA and −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>
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| ===Generalization of quadratic equation===
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| 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.)
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| The symbol
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| :<math>\pm \sqrt {b^2-4ac}</math>
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| in the formula should be understood as "either of the two elements whose square is {{math|''b''<sup>2</sup> − 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}}. Note that 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.
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| ====Characteristic 2====
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| 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
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| :<math>x^{2} + bx + c</math>
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| over a field of characteristic {{math|2}}. If {{math|''b'' {{=}} 0}}, then the solution reduces to extracting a square root, so the solution is
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| :<math>x = \sqrt{c}</math>
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| and note that there is only one root since
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| :<math>-\sqrt{c} = -\sqrt{c} + 2\sqrt{c} = \sqrt{c}.</math>
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| In summary,
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| :<math>\displaystyle x^{2} + c = (x + \sqrt{c})^{2}.</math>
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| See [[quadratic residue]] for more information about extracting square roots in finite fields.
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| In the case that {{math|''b'' ≠ 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
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| :<math>\frac{b}{a}R\left(\frac{ac}{b^2}\right)</math>
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| and
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| :<math>\frac{b}{a}\left(R\left(\frac{ac}{b^2}\right)+1\right).</math>
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| 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>}}.
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| This is a special case of [[Artin–Schreier theory]].
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| ==See also==
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| <div style="-moz-column-count:2; column-count:2">
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| * [[Chakravala method]]
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| *[[Completing the square]]
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| * [[Cubic function]]
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| * [[Fundamental theorem of algebra]]
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| * [[Linear equation]]
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| * [[Parabola]]
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| * [[Periodic points of complex quadratic mappings]]
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| * [[Quadratic function]]
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| * [[Quadratic polynomial]]
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| * [[Quartic function]]
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| * [[Quintic function]]
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| * [[Solving quadratic equations with continued fractions]]
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| </div>
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| ==References==
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| {{reflist|2}}
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| ==External links==
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| * {{springer|title=Quadratic equation|id=p/q076050}}
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| * {{MathWorld|title=Quadratic equations|urlname=QuadraticEquation}}
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| * [http://plus.maths.org/issue29/features/quadratic/index-gifd.html 101 uses of a quadratic equation]
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| * [http://plus.maths.org/issue30/features/quadratic/index-gifd.html 101 uses of a quadratic equation: Part II]
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| * [http://www.gandraxa.com/using_the_quadratic_formula.xml?Var1=3;Var2=1/4;Var3=4 Step-by-step instructions on using the quadratic formula for any input]
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| {{Polynomials}}
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| {{DEFAULTSORT:Quadratic Equation}}
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| [[Category:Elementary algebra]]
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| [[Category:Equations]]
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