Completing the Square

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Description of Levels

Level 1 - Expressions with two terms such as $x^2 + 6x$

Level 2 - Expressions with three terms such as $x^2 + 4x - 7$

Level 3 - The coefficient of the squared term is greater than one such as $2x^2 + 8x - 9$

Level 4 - Use the ability to complete the square to help solve these basic quadratic equations

More Quadratic Equations - Use the ability to complete the square to help solve these more difficult quadratic equations.

Exam Style questions take the skill of completing the square and put it to use solving real problems. Typically problems involve solving equations or describing features of graphs. The questions are in the style of GCSE or IB/A-level exam paper questions and worked solutions are available for Transum subscribers.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don’t yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

Curriculum Reference

See the National Curriculum page for links to related online activities and resources.

Completing The Square

The video above is from the creative and aesthetically mindful Beth.

Completing the square is a technique used to manipulate quadratic expressions into a standard form, which allows for easier factorisation or solution finding.

For example, to complete the square for the quadratic expression $x^2 + 6x + 5$, we follow these steps:

$$ \begin{aligned} x^2 + 6x + 5 &= (x + 3)^2 - 9 + 5 \\ &= (x + 3)^2 - 4 \end{aligned} $$

Therefore, the quadratic expression $x^2 + 6x + 5$ can be written in the standard form $(x + 3)^2 - 4$ after completing the square.

The key formula to complete the square for a quadratic expression of the form $ax^2 + bx + c$ is:

$$ ax^2 + bx + c = a\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a} + c $$

where $a, b,$ and $c$ are constants.

Completing the square is a useful technique in solving quadratic equations and graphing quadratic functions, among other applications.

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly.

Completing the Square

Practise this technique for use in solving quadratic equations and analysing graphs.

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Tran Tunnels

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Description of Levels

Curriculum Reference

Completing The Square