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

are any real numbers, and we assume $a\neq 0$ since otherwise it would not be second order. Some experiments plotting

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

is given by

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

(2.2)

where the magnitude of

determines the width of the parabola, and

provides an arbitrary vertical offset. If

, the parabola has the minimum value

; when

, the parabola reaches a maximum at

(also equal to

). If we can find

in terms of

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

to the general second-order polynomial in Eq. $\,$ (2.1) gives

$\begin{eqnarray*} d &=& a\\ -2 d x_0 &=& b \quad\Rightarrow\quad x_0 = -b/(2a) \\ d x_0^2 + e &=& c \quad\Rightarrow\quad e = c - b^2/(4a). \end{eqnarray*}$

Using these answers, any second-order polynomial 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

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

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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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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See Also

Mathematics of the DFT
Complex Numbers
The Quadratic Formula