# Completing the Square

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

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Solve the following quadratic equations by completing the square. Type in the two roots separated by a comma.

 $$x^2 − 6x + 8 = 0$$ $$x^2 − 5x − 6 = 0$$ $$x^2 − 12x + 35 = 0$$ $$x^2 + 2x − 3 = 0$$ $$x^2 − 13x + 40 = 0$$ $$x^2 + 6x + 8 = 0$$ $$x^2 − 7x + 10 = 0$$ $$x^2 − 2x − 24 = 0$$ $$x^2 − 6x + 5 = 0$$
Check

This is Completing the Square level 4. You can also try:
Level 1 Level 2 Level 3

Finally try some more quadratic equations that are a little more challenging.

## Instructions

Try your best to answer the questions above. Type your answers into the boxes provided leaving no spaces. As you work through the exercise regularly click the "check" button. If you have any wrong answers, do your best to do corrections but if there is anything you don't understand, please ask your teacher for help.

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Learning and understanding Mathematics, at every level, requires learner engagement. Mathematics is not a spectator sport. Sometimes traditional teaching fails to actively involve students. One way to address the problem is through the use of interactive activities and this web site provides many of those. The Go Maths page is an alphabetical list of free activities designed for students in Secondary/High school.

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Tuesday, December 6, 2016

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## Description of Levels

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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.

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## 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.

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