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?

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:

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

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 x 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

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

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.

Harder Examples Complete the square:

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