Completing the Square

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

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Write the following expressions in the completed square form.

This is level 2; Expressions with three terms such as $$x^2 + 4x - 7$$.

 $$x^2+2x+5$$ $$x^2-12x+8$$ $$x^2+2x-10$$ $$x^2-4x+15$$ $$x^2+14x-17$$ $$x^2-10x-21$$ $$x^2+4x-24$$ $$x^2-6x+28$$ $$x^2+6x-30$$
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This is Completing the Square level 2. You can also try:
Level 1 Level 3 Level 4

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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Nine Digits

Arrange the given digits one to nine to make three numbers such that two of them add up to the third. This is a great puzzle for practicing standard pen and paper methods of three digit number addition and subtraction.

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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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