If we take a quadratic expression such as x² + 10x + 25 and write it in the form (x + 5)² we have completed the square since (x + 5)² is known as a perfect square. When completing the square you want to write the quadratic expression in the form (x ± ?)² ± ?. Here is an example to demonstrate the method.

Complete the square on the quadratic expression x² + 8x + 3.

First, we halve the coefficient of x (+4) and put this in the bracket. So the perfect square is (x + 4)².

Next, we subtract the square of the number in the bracket and replace the +3 giving (x + 4)² - 4² + 3

Finally, we simplify the expression. So, x² + 8x + 3 = (x + 4)² - 13

So, I hear you ask, 'How does this help with solving quadratic equations?' When trying to solve quadratic equations which will not factorise completing the square provides us with an alternative method and does not require a calculator unlike using the quadratic formula. If the expression above was the equation x² + 8x + 3 = 0, completing the square enables us to solve it using the balancing method as follows:

x² + 8x + 3 = 0

This is the same as (x + 4)² - 13 = 0

Add 13 to both sides (x + 4)² = 13

Take the square root of both sides: x + 4 = ±√13 (don't forget there are two square roots, one positive and one negative)

Subtract 4 from both sides: x = ±√13 - 4

So there are two solutions x = +√13 - 4 or x = -√13 - 4 and these are best left as surds unless you are told to convert them to decimals.

Here is another example.

Solve the equation x² - 5x - 4 = 0 by completing the square.

Halve the coefficient of x and subtract this value squared: (x - 5/2)² - (5/2)² - 4 = 0

Simplify: (x - 5/2)² - 41/4 = 0

Add 41/4 to both sides: (x - 5/2)² = 41/4

Square root: x - 5/2 = ±√(41/4)

Add 5/2 to both sides: x = 5/2 ±√(41/4) or x = (5 ± √41)/2

So x = (5 + √41)/2 or x = (5 - √41)/2

If there is a coefficient of x² then you must divide the whole equation by this first and then continue as before.

By completing the square, solve: 3x² - 12x + 4 = 0

Divide both sides by 3: x² - 4x + 4/3 = 0

Complete the square: (x - 2)² - (-2)² + 4/3 = 0

Simplify: (x - 2)² - 8/3 = 0

Solve: (x - 2)² = 8/3

x - 2 = ±√(8/3)

x = 2 ±√(8/3)

So x = 2 + √(8/3) or x = 2 - √(8/3)

Now it's over to you.