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Completing the square
Completing the square
is a technique of
elementary algebra
wherein an expression
is replaced by one of the form
Specifically, we have
See
quadratic formula
for the application to the problem of solving the quadratic equation.
Example
Consider the problem of finding this antiderivative:
The denominator is
Adding (10/2)
^{2}
= 25 to
x
^{2}
 10
x
gives a perfect square
x
^{2}
 10
x
+ 25 = (
x
 5)
^{2}
. So we get
Our integral becomes
Completing the square reduces any problem involving a quadratic polynomial to one involving a quadractic polynomial with no 1stdegree term.