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What is completing the square?

Updated: Nov 29, 2017

In this post I’m going to look at completing the square, another fairly tricky topic in GCSE maths. A lot of students learn the procedure for what to do, but don’t quite know why they are doing it. Completing the square is useful in maths because it enables us to solve quadratic equations that do not factorise.


Take a look at this quadratic equation:



It is not possible to solve this equation by factorisation (i.e. by attempting to solve by using the double brackets method), and so there are two other methods we can use. The first is to use the quadratic formula:


The other method, which I am going to describe below, is to complete the square.


Using the same quadratic equation, take out an x, and add it to half of the coefficient of the second term:



Then, square:



At this stage, it is useful to know what exactly we have done. If you expand the above, you get:



Compare this to the original quadratic equation:



The bit I have highlighted is the same as the following equation with the fraction removed:



Therefore:



All we then have to do is not forget to add the 2. We have then converted the original quadratic expression into an alternative form that only involves one x:



With only one x, it is now possible to solve:



Have a look at this video which goes through similar completing the square examples:





Where does the quadratic formula come from?


As an interesting aside, the quadratic formula can also be derived by completing the square. This is beyond GCSE maths, but can you spot where completing the square has been used? You first start with the generic quadratic equation:










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