|
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
|