Editing Quadratic equation (section)

====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>