# The Quadratic Formula

## GUIDE: Mathematics of the Discrete Fourier Transform (DFT) - Julius O. Smith III. The Quadratic Formula

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# The Quadratic Formula

The general second-order polynomial is where the coefficients are any real numbers, and we assume 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 where determines the width and provides an arbitrary vertical offset. 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 , we get Equating coefficients of like powers of gives Using these answers, any second-order polynomial can be rewritten as a scaled, translated parabola In this form, the roots are easily found by solving 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 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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