Understand How to Complete the Square in Complex Quadratics

Completing the square (ax² + bx + c)

We have shown in previous worksheets for the other levels, that we can complete the square for a quadratic. For example: x² + 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 ax² + bx + c

 

What is the general form?

When we complete the square for a quadratic of the form ax² + bx + c, we use this: d(x ± e)² ± f

 

Example:

Complete the square for 3x² – 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:

3x² – 24x + 5 = 3{x² – 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.

x² – 8x = (x + 4)² – 16

 

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

3{x² – 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.