Quadratic Expressions
Complete Squares
Exercise
Write each of the following as a complete square.
(a) x  10x + 25
(b) z + 8z + 16
(c) w  w + 1/4
(d) y + 5y + 25/4
Solution
(a) Comparing x  2ax + a with x  10x + 25 we see that 2a = 10 , and a = 25 . Thus a = 5 is the solution. It may
easily be checked that
(x  5) = x  10x + 25 .
(b) Comparing z  2az + a with z + 8z + 16, it follows that 2a = 8 , and a = 16 . In this case a = 4. Checking
(z  ( 4)) = (z + 4) = z + 8z + 16 .
(c) Comparing w  2aw + a with w  w + 1/4, we must have 2a = 1 , and a = 4 . In this case a = 1/2 , and
It is now easy to check that
(d) Comparing y  2ay + a with y + 5y + 25/4 we must have 2a = 5 , and a = 25/4 . From this it must follow that a =
5/2 .
Note that
It is now easy to check that
Quiz
Which of the following quadratic expressions is a complete
square?
(a) z + 3 z  9 /4 (b) z + 3 z + 9 /2 (c) z  3 z + 9 /4 (d) z  3 z + 9 /2
Solution
Comparing z  3z + 9/4 with the general form of a
complete square (z  a) = z  2az + a
if 2a = 3 then ,
and
Now it is easily checked that
