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Completing the SquarePractise this technique for use in solving quadratic equations and analysing graphs. |
Write the following expressions in the completed square form.
This is level 1; Expressions with two terms such as \(x^2 + 6x\).
InstructionsTry 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. When you have got all of the questions correct you may want to print out this page and paste it into your exercise book. If you keep your work in an ePortfolio you could take a screen shot of your answers and paste that into your Maths file. |
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Mathematicians are not the people who find Maths easy; they are the people who enjoy how mystifying, puzzling and hard it is. Are you a mathematician? Comment recorded on the 14 October 'Starter of the Day' page by Inger Kisby, Herts and Essex High School: "Just a quick note to say that we use a lot of your starters. It is lovely to have so many different ideas to start a lesson with. Thank you very much and keep up the good work." Comment recorded on the 25 June 'Starter of the Day' page by Inger.kisby@herts and essex.herts.sch.uk, : "We all love your starters. It is so good to have such a collection. We use them for all age groups and abilities. Have particularly enjoyed KIM's game, as we have not used that for Mathematics before. Keep up the good work and thank you very much |
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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.
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.
See the National Curriculum page for links to related online activities and resources.
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.
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.
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Colleen Young, Twitter
Tuesday, December 6, 2016