Editing Quadratic equation (section)

==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]].