How do you complete the square?

'Completing the square' is quite a tricky concept to some people, and I honestly didnt grasp it the first time i was taught it. But once explained thoroughly, it becomes easier to use.

'Completing the square' is a method for solving quadratic equations, when an equation cannot be easily factorised. In fact, the quadratic formula you will see in formulae books, is proven by 'completing the square' of the following equation: ax2​ + bx + c = 0.

I am going to explain the 'completing the square' method by using the following example.

I will use the equation x2​ - 6x + 5 =0.

Those of you who are quite eagle eyed will notice that you can easily factorise this equation and see that the solutions are x=1 and x=5, but i want to show you how to find these solutions by 'completing the square'.

The first step involves putting the x into a bracket with a squared on the outside. To do this you need to look at the first two terms: x2​ - 6x.

After the first step the equation should look like this: (x-3)2​ - 9 + 5 = 0. I will explain why you do this. When you look at x2​ -6x you need to ask yourself the follwing question: what expression in x can i square to get these two terms?

By asking yourself this question you might notice that if you are going to square an expression, then the number within the expression should be half the number of x's in your original question. This is how x2 - 6x becomes (x-3)2​.

In reality, the x2 - 6x becomes (x-3)2​ - 9. The reason this happens lies in the expansion of (x+3)2​. When we expand the bracket we get x2​ - 6x + 9, which is not what we want as it is 9 more than the expression we want. This is why the -9 appears to fix this problem.

Now we have (x - 3)2​ - 9 + 5 = 0. We can condense this down to (x - 3)2​ - 4 =0. The next step is to isolate the squared bracket; this means writing the equation as (x - 3)2​ = 4.

As you can see, the equation is looking a lot nicer know, and the next step is to square root both sides of the equation. This will make two equations as the square root of 4 is both 2 and -2.

We know have x - 3 = 2 and x - 3 = -2. Adding 3 to both sides of both equations gives the required results of x = 5 and x = 1.

And there you have it, completing the square is best done when the coefficient of x2​ is 1, but if you have a different coefficient, just divide all the numbers in the equation by the coefficient and then complete the square.

Alex W. A Level Maths tutor, GCSE Maths tutor, A Level Further Mathem...

3 weeks ago

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