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A general Series for expressing the Root of any Quadratick Equation. Any Quadratick Equation being reduc’d to this Form <math>xx - mqx + my = 0</math… - Brook Taylor
" "A general Series for expressing the Root of any Quadratick Equation.
Any Quadratick Equation being reduc’d to this Form <math>xx - mqx + my = 0</math>, the Root <math>x</math> will be exprest by this Series of Terms. <math> x = \frac {y}{q} + A \times \frac{1}{\frac{mq^2}{y} -2} + B \times \frac {1}{a^2 - 2} + C \times \frac {1}{b^2 - 2} +D \times \frac {1}{c^2 - 2}</math> etc. Which must be thus interpreted.
1. ...A, B, C, etc. stand for the whole terms with their Signs, preceding those wherein they are found, as <math>B = A \times \frac {1}{\frac{mq^2}{y} - 2}</math>
2. ...<math>a, b, c,</math> etc. ...are equal to the whole Divisors of the Fraction in the Terms immediately preceding; thus <math>b = a^2 - 2</math>.
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About Brook Taylor
Brook Taylor (18 August 1685 – 29 December 1731) was an English mathematician and secretary of the Royal Society of London, most famous for Taylor's theorem and the Taylor series.
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Hence if <math>y</math> be the Root of any Expression formed of <math>y</math> and known Quantities, and supposed equal to nothing, and <math>z</math> be a part of <math>y</math>, and <math>x</math> be formed of <math>z</math> and the known Quantities, in the same manner as the Expression made equal to nothing is formed of <math>y</math>; and let <math>y</math> be equal to <math>z + v</math>; the difference <math>v</math> will be found by Extracting the Root of this expression <math>x + \frac {\dot{x}v}{1} + \frac {\ddot{x} v^2}{1 \cdot 2} + </math> ... etc. <math>= 0</math>.
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