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Understand How to Complete the Square in Complex Quadratics

In this worksheet, students will practise completing the square for quadratics.

'Understand How to Complete the Square in Complex Quadratics' worksheet

Key stage:  KS 4

Year:  GCSE

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, OCR, Pearson Edexcel,

Curriculum topic:   Algebra

Curriculum subtopic:   Notation, Vocabulary and Manipulation Algebraic Expressions

Difficulty level:  

Worksheet Overview

Completing the square (ax+ 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



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.

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