The general second-order (real) polynomial is

 (2.1)

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

 (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 at ; 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

 (2.3)

Equating coefficients of like powers of to the general second-order polynomial in Eq.(2.1) 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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Complex Roots
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Factoring a Polynomial