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Completing the square This is an important algebraic technique that has important applications in the study of the properties of parabolic graphs. There will be a question on completing the square in the Unit 1 assessment. Remember that you MUST pass this assessment if you want to complete the N5 course The key idea How do I complete the square? What is completing the square for?

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The key idea Expressions involving x 2 are called quadratic expressions, and some quadratics can be written as perfect squares: The factorisation patterns that you need to know are:

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Make sure you can answer this before continuing This expression doesn’t fit the pattern because it isn’t a perfect square What would you have to do to this expression to MAKE it into a perfect square? You would have to add 16 to it. Adding 16 to the expression lets you write the quadratic as a perfect square. This is what is meant by completing the square

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How do I complete the square? If the expression involves x 2 (and not 2x 2 or 3x 2 etc.) start by halving the coefficient of x: For example - complete the square for this expression Half of 10 is 5 so the perfect square that produces x 2 + 10x is (x + 5) 2 but our expression doesn’t have the 25. So we do this: Be careful here, if we just add 25 then the new expression will not be equal to the first expression so you must ADD and SUBTRACT the 25

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Completing the square practice questions Complete the square for these 3 examples on paper then check your answers (remember halve the x coefficient)

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But what if it’s 2x 2 or -x 2 ? If the quadratic has a multiplier of x 2 or if the square term is negative then you have to take out a common factor first. Study these 2 examples then try some for yourself on the next page Complete the square of the expression inside the bracket. At the end, always multiply through by the common factor.

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Harder Examples Complete the square:

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What is completing the square for? There are two important ideas bundled up in this technique. Algebraically, it allows you to solve equations involving an x 2 term by giving you a method for solving any quadratic equation: It lets you make x the subject of any quadratic equation (see unit 2 for more information) Secondly, it lets you exploit an important property of (real) squares namely: Squares can never be negative In unit 2 you will see how you can use this to locate maxima and minima of quadratic graphs. Remember, in unit 1 you just need to use the process

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