**Completing the square (ax ^{2 }+ bx + c)**

We have shown in previous worksheets for the other levels, that we can complete the square for a quadratic. For example: x^{2} + 6x – 2 = (x + 3)^{2} - 11

This, however, will only work if the coefficient of **x is 1**. This activity will look at how we deal with quadratics of the form ax^{2} + bx + c

**What is the general form?**

When we complete the square for a quadratic of the form ax^{2} + bx + c, we use this: d(x ± e)^{2} ± f

**Example: **

**Complete the square for 3x ^{2} – 24x + 5**

The trick with this, is to convert it into a form that we can deal with (i.e. the number in front of x is 1).

To do this, we** factorise** the **first two terms:**

3x^{2} – 24x + 5 = 3{x^{2} – 8x} + 5

We can now complete the square for the expression in the bracket. If you aren’t sure about this, take a look at activity 7942 on completing the square.

x^{2} – 8x = (x + 4)^{2} – 16

If we substitute this back into our factorised quadratic, we get:

3{x^{2} – 8x} + 5 = 3{(x + 4)^{2} – 16} + 5

Our penultimate step is to multiply by the 3 we factorised out earlier:

3{(x + 4)^{2} – 16} + 5 = 3(x + 4)^{2} – 48 + 5

And the final step is to collect the like terms:

3(x + 4)^{2} – 48 + 5 = 3(x + 4)^{2} – 43

There's lots to remember here, so have another look through the example before attempting the questions.