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 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
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.