# 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

x^{2} + bx = (x + b/2)^{2} - (b/2) ^{2}

## 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

## 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

- Step by Step Solver Calculator to Complete the Square
- Find Vertex and Intercepts of Quadratic Functions - Calculator: An applet to solve calculate the vertex and x and y intercepts of the graph of a quadratic function.
- Quadratic functions (general form).
- Quadratic Functions - Problem (1).
- Quadratic functions (standard form).
- Graphing quadratic functions.