The Quadratic Formula
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.
Next Section: Complex RootsPrevious Section: Factoring a Polynomial