Write x^2 – 10x + 12 in the form (x – a)^2 + b , where a and b are integers.

This is a past exam question from a real GCSE paper, and is an example of "completing the square." I will show you 4 steps for answering this question. ¬¬¬ The first step is to check whether the x^2 term has a 'coefficient' different to 1, (is there a number 'attached to' to x^2, for example 2x^2.) In this case there is no coefficient, so we move to the second step. ¬¬¬ The second step is to find 'a' by looking at the coefficient of the x term, which in this case is 10. (The number 'attached to' to the x.) Now we half this number. Half of 10 is 5. 10 / 2 = 5 We have found a! a = 5 ¬¬¬ Third step: square the number we found for a, (here 5,) and subtract this from our "constant term", (the number with no x or x^2 attached to it.) Here this number is 12. 5^2 = 25 12 - 25 = -13 so 'b' is -13. ¬¬¬ We can put this number where the 'b' is in the form given. ¬¬¬ Final step: at last we can put everything together. We found that a = 5 and b = -13, so put these numbers into the form given in the question: (x -5)^2 -13 ¬¬¬ This is the final answer. [You can check this in an exam by expanding the brackets of your answer to make sure you get the original expression.] ¬¬¬

This was a difficult question and most students struggled with this in the exam, according to the Examiner's report. However, this type of question can be straightforward if you learn the method for "completing the square"!