Quadratic equation
Trigonometric solution

In the days before calculators, people would use mathematical tables —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. 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,

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

and then multiplying through by cos (θ) / c, we obtain

Introducing functions of 2 θ and rearranging, we obtain

where the subscripts n and p correspond, respectively, to the use of a negative or positive sign in equation. Substituting the two values of θ or θ found from equations or into gives the required roots of. Complex roots occur in the solution based on equation if the absolute value of sin 2 θ 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. Calculating complex roots would require using a different trigonometric form.

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: 4.16130 x 2 + 9.15933 x − 11.4207 = 0 {\displaystyle 4.16130x^{2}+9.15933x-11.4207=0}
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