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**, we need to write the quadratic expression in the form **(x ± ?)² ± ?**

Here is an example to demonstrate this method.

**Complete the square for 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 which 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 this 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 possible 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**

These are best left as **surds** unless you are told to convert them to decimals.

Here is another example to follow through.

**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 we 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 to solve some quadratic equations using the method of completing the square.

These methods can be long and tricky to remember, so it is a good idea to have a pen and paper handy.

If you write out your method for each question, you can compare your working with what our maths teacher has written.