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The Quadratic Formula

The general second-order (real) polynomial is

$\displaystyle p(x) \isdef a x^2 + b x + c \protect$ (2.1)

where the coefficients $ a,b,c$ are any real numbers, and we assume $ a\neq 0$ since otherwise it would not be second order. Some experiments plotting $ p(x)$ for different values of the coefficients leads one to guess that the curve is always a scaled and translated parabola. The canonical parabola centered at $ x=x_0$ is given by

$\displaystyle y(x) = d\cdot (x-x_0)^2 + e \protect$ (2.2)

where the magnitude of $ d$ determines the width of the parabola, and $ e$ provides an arbitrary vertical offset. If $ d>0$ , the parabola has the minimum value $ e$ at $ x=x_0$ ; when $ d<0$ , the parabola reaches a maximum at $ x=x_0$ (also equal to $ e$ ). If we can find $ d,e,x_0$ in terms of $ a,b,c$ for any quadratic polynomial, then we can easily factor the polynomial. This is called completing the square. Multiplying out the right-hand side of Eq.(2.2) above, we get

$\displaystyle y(x) = d(x-x_0)^2 + e = d x^2 -2 d x_0 x + d x_0^2 + e. \protect$ (2.3)

Equating coefficients of like powers of $ x$ to the general second-order polynomial in Eq.(2.1) gives

d &=& a\\
-2 d x_0 &=& b \quad\Rightarrow\quad x_0 = -b/(2a) \\
d x_0^2 + e &=& c \quad\Rightarrow\quad e = c - b^2/(4a).

Using these answers, any second-order polynomial $ p(x) = a x^2 + b x + c$ can be rewritten as a scaled, translated parabola

$\displaystyle p(x) = a\left(x+\frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right).

In this form, the roots are easily found by solving $ p(x)=0$ to get

$\displaystyle \zbox {x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.}

This is the general quadratic formula. It was obtained by simple algebraic manipulation of the original polynomial. There is only one ``catch.'' What happens when $ b^2 - 4ac$ is negative? This introduces the square root of a negative number which we could insist ``does not exist.'' Alternatively, we could invent complex numbers to accommodate it.

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University