Quadratic Expressions
Completing the Square
Not every quadratic is a complete square but it is possible to
write ALL quadratics as a complete square plus a number. This is
the process known as COMPLETING THE SQUARE. This is an extremely
useful algebraic procedure with many applications.
As an example, let us take the quadratic expression x  4 x + 5 . From the beginning of Section
3 we note that
x 4 x + 4 = ( x  2) .
Since x  4 x + 5 = ( x  4 x + 4) + 1 ,
the expression may be written as
x  4 x + 5 = ( x  2) + 1 ,
and we have completed the square on the quadratic.
Example 1
Complete the square on each of the following quadratic
expressions.
( a) x + 6 x + 11 ,
( b) x + 4 x + 3 ,
( c) 2 x + 8 x + 4 ,
( d) x  2 ax + a + b .
Solution
(a) Since x + 6 x + 11 = ( x + 6 x + 9) + 2 , the expression may be
written x + 6 x + 11 = ( x + 3) + 2 .
(b) Here we note that x + 4 x + 3 = ( x + 4 x + 4)  1 .
Now we may use the fact that ( x + 2) = ( x + 4 x + 4) , to obtain
x + 4 x + 3 = ( x + 4 x + 4)  1 = ( x + 2)  1 .
(c) Since 2 x + 8 x + 4 = 2( x + 4 x + 2) = 2( { x + 4 x + 4}  2) ,
we have 2 x + 8 x + 4 = 2( { x + 2 }  2) = 2( x + 2)  4 .
(d) Here x  2ax + a + b = ( x  a) + b .
