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