This is a tutorial on completing the square of quadratic expressions.
The idea of completing the square stems from the following result
x^{2} + bx = (x + b/2)^{2}  (b/2) ^{2}
The above could be applied to any quadratic expression.
Examples with Detailed Solutions
Example 1
x^{2} + 4x = (x + 4/2)^{2}  (4/2) ^{2} = (x + 2)^{2}  4
Example 2
x^{2} + 2x + 5 = (x + 2/2)^{2}  (2/2) ^{2} + 5 = (x + 1)^{2}  1 + 5 = (x + 1)^{2} + 4
When completing the square when the leading coefficient is not equal to 1, we factor out the leading coefficient and work inside the brackets.
Example 3
2x^{2}  12x = 2[ x^{2}  6x ]
= 2[ (x + (6/2) )^{2}  (6/2)^{2} ]
= 2[ (x  3 )^{2}  9 ]
= 2(x  3 )^{2}  18
Example 4
x^{2}  10x =  [ x^{2} + 10x ]
= [ (x + 10/2)^{2}  (10/2)^{2} ]
=  [ (x + 5)^{2}  25 ]
= (x + 5)^{2} + 25
Example 5
2x^{2}  3x = 2 [ x^{2} + (3/2) x ]
= 2 [ (x + (3/4))^{2}  (3/4)^{2} ]
= 2 (x + (3/4))^{2} + 9/8
When completing the square, leave any constant term outside the brackets.
Example 6
3x^{2} + 2x + 2 = 3 [ x^{2}  (2/3) x ] + 2
= 3 [ (x + (2/6))^{2}  (2/6)^{2} ] + 2
= 3 [ (x  1/3))^{2}  (1/3)^{2} ] + 2
= 3 [ (x  1/3))^{2}  1/9 ] + 2
= 3 (x  1/3))^{2} + 1/3 + 2
= 3 (x  1/3))^{2} + 7/3
Exercise:Complete the square for the following quadratic expressions (answers given below)
Exercises
Complete the square for the following quadratic expressions (answers given below)
A) x^{2} + 4x
B) x^{2} + 6x  3
C) x^{2}  4x + 2
D) 2x^{2}  5x  5
Solutions to Exercises Above
A) x^{2} + 4x = (x + 2)^{2}  4
B) x^{2} + 6x  3 = (x + 3)^{2}  12
C) x^{2}  4x + 2 = (x + 2)^{2} + 6
D) 2x^{2}  5x  5 = 2(x + 5/4)^{2}  15/8
More references and links on the quadratic functions in this website
