Completing the square - Higher

This is another way to solve a quadratic equation if the equation will not factorise.

It is often convenient to write an algebraic expression as a square plus another term. The other term is found by dividing the coefficient of x by 2, and squaring it.

Any quadratic equation can be rearranged so that it can be solved in this way.

Have a look at this example.

Example 1

Rewrite x² + 6x as a square plus another term.

The coeffient of x is 6. Dividing 6 by 2 and squaring it gives 9.

x² + 6x = (x² + 6x + 9) - 9

= (x + 3)² - 9

Example 2

We have seen in the previous example that x² + 6x = (x + 3)² - 9

So work out x² + 6x - 2

x² + 6x - 2 = ( x² + 6x + 9 ) - 9 - 2 = (x + 3)² - 11

Now try one for yourself.

Example 3

Rewrite 2x² + 20x + 3

Rewrite to get x² on its own.

2( x² + 10x ) + 3

• The coefficient of x is 10. Divide 10 by 2, and square to get 25.
• = 2 ( ( x + 5)² - 25) + 3
• = 2 (x + 5)² - 50 + 3
• = 2 (x + 5)² - 47