The Quadratic Formula

GUIDE: Mathematics of the Discrete Fourier Transform (DFT). The Quadratic Formula

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The Quadratic Formula

The general second-order polynomial is

\

where the coefficients$a,b,c$ are any real numbers, and we assume $a\ 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

\

where $d$ determines the width and $e$ provides an arbitrary vertical offset. 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 $y(x)$, we get

\

Equating coefficients of like powers of $x$ gives

\



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

\

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

\

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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