Completing the Square of Quadratic Expressions

This is a tutorial on completing the square of quadratic expressions.

The idea of completing the square stems from the following result

x2 + bx = (x + b/2)2 - (b/2) 2

The above could be applied to any quadratic expression such as

Example 1

x2 + 4x = (x + 4/2)2 - (4/2) 2 = (x + 2)2 - 4

Example 2

x2 + 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

2x2 - 12x = 2[ x2 - 6x ]

= 2[ (x + (-6/2) )2 - (-6/2)2 ]

= 2[ (x - 3 )2 - 9 ]

= 2(x - 3 )2 - 18

Example 4

-x2 - 10x = - [ x2 + 10x ]

= -[ (x + 10/2)2 - (10/2)2 ]

= - [ (x + 5)2 - 25 ]

= -(x + 5)2 + 25

Example 5

-2x2 - 3x = -2 [ x2 + (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

-3x2 + 2x + 2 = -3 [ x2 - (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 follwoing quadratic expressions

(answers given below)

A) x2 + 4x

B) x2 + 6x - 3

C) -x2 - 4x + 2

D) -2x2 - 5x - 5


Solutions to Exercises Above :


A) x2 + 4x = (x + 2)2 - 4

B) x2 + 6x - 3 = (x + 3)2 - 12

C) -x2 - 4x + 2 = -(x + 2)2 + 6

D) -2x2 - 5x - 5 = -2(x + 5/4)2 - 15/8




More references and links on the quadratic functions in this website.