# Completing the square

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## Completing the square

A quadratic equation is an equation where the highest power of x is x2., so it is an equation of the form ax2 + bx + c = 0. There are various methods of solving quadratic equations, as shown below.

NOTE: If x2 = 36, then x = +6 or -6 (since squaring either of these numbers will give 36). However, if we write √36, we usually mean +6 .

### Completing the Square

9 and 25 can be written as 32 and 52 whereas 7 and 11 cannot be written as the square of another exact number. 9 and 25 are called perfect squares. Another example is (9/4) = (3/2)2. In a similar way, x2 + 2x + 1 = (x + 1)2.

To make x2 + 6x into a perfect square, we add (62/4) = 9. The resulting expression, x2 + 6x + 9 = (x + 3)2 and so is a perfect square. The process of making something into a perfect square is known as completing the square. To complete the square in this way, we take the number before the x, square it, and divide it by 4. This technique can be used to solve quadratic equations, as demonstrated in the following example.

#### Example

Solve x2 - 6x + 2 = 0 by completing the square
x2 - 6x = -2
[To complete the square on the LHS (left hand side), we must add 62/4 = 9. We must, of course, do this to the RHS also].
x2 - 6x + 9 = 7
(x - 3)2 = 7
[Now take the square root of each side]
x - 3 = ±2.646     (the square root of 7 is +2.646 or -2.646)
x = 5.646 or 0.354

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Let's complete the square in the general case: ax2 + bx + c

Take out a factor of a:
a [ x2 + (b/a)x + (c/a) ]
a [ [x + (b/2a)]2 + (c/a) - (b2/4a2) ]

Hence if ax2 + bx + c = 0,
[x + (b/2a) ]2 = (b2/4a2) - (c/a)
=b2 - 4ac
4a2

Now if we take the square root of both sides and simplify, we get the quadratic formula:

#### Example

Solve 3x2 + 5x - 8 = 0

x = -5 ± √( 52 - 4×3×(-8))
6
= -5 ± √(25 + 96)
6
= -5 ± √ (121)
6
= -5 + 11       or        -5 - 11
6                         6

x = 1 or -2.67

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## Factorising

#### Example

Solve x2 + 2x - 8 = 0
(x - 2)(x + 4) = 0
either x - 2 = 0 or x + 4 = 0
x = 2 or x = - 4

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