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 .