MYTUTOR SUBJECT ANSWERS

179 views

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...

5 months ago

Answered by Alex, a GCSE Maths tutor with MyTutor


Still stuck? Get one-to-one help from a personally interviewed subject specialist

458 SUBJECT SPECIALISTS

£22 /hr

Daniel R.

Degree: Mathematics (Bachelors) - Durham University

Subjects offered: Maths, Physics+ 1 more

Maths
Physics
Further Mathematics

“About me I’m a first year student studying Maths with European Studies at Durham. I have recently taken my A levels, achieving A* in Maths and Further Maths, so I am familiar with the course content and what the examiners are looking ...”

£18 /hr

George S.

Degree: Physics (Masters) - Birmingham University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
.PAT.

“About Me: I am just entering my first year to study Physics at University of Birmingham, I've always enjoyed anything maths based hence my A-level choices, I hope I can help you through your work whether it's GCSE or A-level and maybe...”

£18 /hr

Matthew H.

Degree: Physics (Masters) - Liverpool University

Subjects offered: Maths, Physics+ 3 more

Maths
Physics
History
Chemistry
-Personal Statements-

“About Me  I am studying physics at the University of Liverpool and am now in my second year having passed with a first in all my modules last year. I really love all things science and have a real drive to pass on what I know to other...”

About the author

£18 /hr

Alex W.

Degree: Mathematics (Masters) - Warwick University

Subjects offered: Maths, Physics+ 1 more

Maths
Physics
Further Mathematics

“Hi i'm Alex. I am currently a first year Maths student at Warwick University, and can tutor you in Maths, Further Maths and Physics at both GCSE and A-Level. I have had plenty of teaching experience before so I can hopefully help you ...”

MyTutor guarantee

You may also like...

Posts by Alex

How do you complete the square?

How do you use derivatives to categorise stationary points?

What are polar coordinates?

What are the main causes of energy loss in a transformer?

Other GCSE Maths questions

How do I find f'(x) for f(x)=4x^3+x^2+5x+8?

What is the highest common factor of 24 and 90?

How do I use Pythagorus' Theorum?

Solve the simultaneous equations 3x + 4y = 17 and 4x + y = 14

View GCSE Maths tutors

Cookies:

We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok