Completing the Square

Completing the Square

Article Contents

What is Completing the Square
Basic Procedure
Deriving the Formula
Examples
Quadratic Equations
Interactive Exercises

1. What is Completing the Square {#what-is-completing}

Completing the square is a technique that converts a quadratic expression to the form

(x + p)^{2} + q

.

For example:

x^{2} + 6 x + 5 = (x + 3)^{2} - 4

Why is it Useful?

Solving quadratic equations
Finding the vertex of a parabola
Integration (later in mathematics)

2. Basic Procedure {#basic-procedure}

For an expression $x^{2} + b x$ :

The coefficient of $x$ is $b$
Half of $b$ is $\frac{b}{2}$
Add and subtract $(\frac{b}{2})^{2}$

x^{2} + b x = x^{2} + b x + (\frac{b}{2})^{2} - (\frac{b}{2})^{2} = (x + \frac{b}{2})^{2} - (\frac{b}{2})^{2}

3. Deriving the Formula {#deriving}

Example $x^{2} + 6 x$

Coefficient $b = 6$

\frac{b}{2} = 3, (\frac{b}{2})^{2} = 9

x^{2} + 6 x + 9 - 9 = (x + 3)^{2} - 9

4. Examples {#examples}

Example 1: $x^{2} + 4 x + 1$

x^{2} + 4 x + 1 = (x^{2} + 4 x + 4) - 4 + 1 = (x + 2)^{2} - 3

Example 2: $x^{2} - 8 x + 5$

x^{2} - 8 x + 5 = (x^{2} - 8 x + 16) - 16 + 5 = (x - 4)^{2} - 11

Example 3: $2 x^{2} + 8 x + 3$

First factor out 2:

2 (x^{2} + 4 x) + 3 = 2 [(x^{2} + 4 x + 4) - 4] + 3 = 2 (x + 2)^{2} - 8 + 3 = 2 (x + 2)^{2} - 5

5. Quadratic Equations {#quadratic-equations}

Solving $x^{2} + 6 x + 5 = 0$

By completing the square:

x^{2} + 6 x + 9 = - 5 + 9

(x + 3)^{2} = 4

x + 3 = \pm 2

x = - 1 or x = - 5

6. Interactive Exercises {#interactive-exercises}

Practice completing the square:

Summary

Expression	Completed Form
$x^{2} + b x$	$(x + \frac{b}{2})^{2} - (\frac{b}{2})^{2}$
$x^{2} + 6 x + 5$	$(x + 3)^{2} - 4$
$x^{2} - 8 x + 5$	$(x - 4)^{2} - 11$

Difference of Squares – related technique
Introduction to Algebra – basics
Perfect Cube – advanced formulas