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The Quadratic Formula
The general secondorder (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 righthand side of Eq.
(2.2) above, we get

(2.3) 
Equating coefficients of like powers of
to the general secondorder
polynomial in Eq.
(2.1) gives
Using these answers, any secondorder 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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