Look at the following quadratic expression:
x - 4x + 4 = (x - 2)(x - 2) = (x - 2) .
For the obvious reason, this expression is called a complete
square. Quadratic expressions which can be factored into
a complete square are useful in many situations. They have a
particularly simple structure and it is important to be able to
recognise such factorizations.
Show that the following quadratic expressions are complete
(a) x + 6 x + 9 , (b) x + 4 x + 4 , (c) x - 2 x + 1 , (d) x - 2 ax + a 2 .
In each of these cases it is easy to check the following.
(a) x + 6 x + 9 = ( x + 3) ,
(b) x + 4 x + 4 = ( x + 2) ,
(c) x - 2 x + 1 = ( x - 1) ,
(d) x - 2 ax + a = ( x - a) .
The last example, x - 2ax + a = (x - a), is a general case and it may be used to
find perfect squares for any given example. It may be usefully
employed in finding solutions to the following exercises.