Completing the square - I don't get it...

Hey there, I've been looking through my maths work that we've done so far (C1 obviously) and I'm up to completing the square and I just don't get what to do, I can't work out what I'm doing from my own notes. I looked at a textbook and still don't really get it.

Theres a few different types of questions from the textbook, the questions sort of build up as you go through it (core maths for advanced level)

So the first set say: Add a number to each expression so that the result contains a perfect square as a factor. (I don't really know what I'm supposed to be doing)
Heres an e.g. of a question: x^2 - 4x, another is 2x^2 - 4x, and so on.

But here are by the looks of it the main type of question: Solve the equations by completing the square, giving the solutions are in surd form:
I'll try and give a variation of questions...

1. X^2 + 8x = 1
2. 2X^2 - x - 2 = 0
3. 2x^2 + 4x = 7

But yeah, if anyone could show me one of those examples with all the steps explained I'd be grateful, because I really don't know what I'm doing! What is actually meant by completing the square?
The answers are in the back of the textbook but of course there are no steps so its pretty useless for me to write the answer straight down.
Thanks a lot

Reply 1

Ewan

15

1. X^2 + 8x = 1
X^2 + 8x - 1 = 0 [-1 from both sides]
[[x + 4]^2 -16] - 1 = 0 [factorising X^2 + 8x and subtracting the remainder created by 4*4]
[x+4]^2 - 17 = 0 Done [completing the fraction.. ie -16-1 to get the final answer]

Basically.. look at this part: X^2 + 8x and factorise it, ie [x+4]^2, but then that creates a +16 that you don't want (since you must keep it the same as the original equation]

For question two... look at 2X^2 - x, and factorise it ie [x-1/2]^2... see if you can do the rest. [you get a +1/4 but you need -2 so you will have to subtract something]

Reply 2

scraceus999

11

1. x^2+8x=1.

first put the 1 on the end the other side of equation, so x^2+8x-1=0.

then put (x^2+the coefficiant of x divided by 2), so (x^2-4x)^2.

then double coefficient of x in brackets and change sign, placing outside bracket,
so (x^2-4x)^2+8, and then -1=0 on the end (from the top equation).

then simplify. (x^2-4x)^2+7=0.

take the coefficiant of x in the bracket and change the sign to get the first value of x, so it is 4.

then take the number outside the bracket, so 7.

x= 4 or 7.

Reply 3

scraceus999

11

okay ignore my comment, i forgot a crucial stage!!! whoopsie.

Reply 4

Ewan

15

Oh and... [x+4]^2 - 17 = 0

To find x... you just solve in a regular way... [x+4]^2 = 17 etc

Reply 5

Icy_Mikki

13

The Song
This is how we complete the square
Complete the square
Complete the square
This is how we complete the square
Of a simple quadratic function

Half the coefficient of x
efficient of x
efficient of x
Half the coefficient of x
Then subtract its square from the constant.

What it actually means
So, always arrange the equation into the form like 2x^2 + 5x + 13 = 0
Then divide everything by the coefficient of x^2, to get 2(x^2 + 5/2x + 13/2) = 0
Then complete the square on the (x^2 + 5/2x + 13/2) part

"Half the coefficient of x"
Half of 5/2 is 5/4, right?
Then put this 5/4 in a bracket like so: (x-5/4)^2. Step one done.

"Then subtract its square from the constant"
i.e. subtract (5/4)^2 from the original constant, which was 13/2- thus leaving us with 13/2 - (5/4)^2. Step 2 done.

Then we add the two parts formed together.
So x^2 + 5/2x + 13/2 = (x-5/4)^2 + (13/2 - (5/4)^2)

Since the equation was originally doubled, then
2x^2 + 5x + 13
= 2(x^2 + 5/2x + 13/2)
= 2((x-5/4)^2 + (13/2 - (5/4)^2))
= 2(x-5/4)^2 + 2(13/2 - (25/16))
= 2(x-5/4)^2 + 13 - (25/8)
= 2(x-5/4)^2 + 79/8

And we've just completed a hard "complete the square" question! :smile:

Reply 6

scraceus999

11

Icy_Mikki

The Song
This is how we complete the square
Complete the square
Complete the square
This is how we complete the square
Of a simple quadratic function

Half the coefficient of x
efficient of x
efficient of x
Half the coefficient of x
Then subtract its square from the constant.

What it actually means

So, always arrange the equation into the form like 2x^2 + 5x + 13 = 0
Then divide everything by the coefficient of x^2, to get 2(x^2 + 5/2x + 13/2) = 0
Then complete the square on the (x^2 + 5/2x + 13/2) part

"Half the coefficient of x"
Half of 5/2 is 5/4, right?
Then put this 5/4 in a bracket like so: (x-5/4)^2. Step one done.

"Then subtract its square from the constant"
i.e. subtract (5/4)^2 from the original constant, which was 13/2- thus leaving us with (5/4)^2 - 13/2. Step 2 done.

So x^2 + 5/2x + 13/2 = (x-5/4)^2 + (13/2 - (5/4)^2)

Since the equation was originally doubled, then
2x^2 + 5x + 13
= 2(x^2 + 5/2x + 13/2)
= 2((x-5/4)^2 + (13/2 - (5/4)^2))
= 2(x-5/4)^2 + 2(13/2 - (25/16))
= 2(x-5/4)^2 + 13 - (25/8)
= 2(x-5/4)^2 + 79/8

And we've just completed a had "complete the square" question! :smile:

how cool i cant believe some people get to learn maths through songs :frown:

Reply 7

Diaz

12

scraceus999

okay ignore my comment, i forgot a crucial stage!!! whoopsie.

Lol exactly what I was thinking!

Reply 8

danielharris627

OP

13

Wow, thats actually pretty easy when broken down! Thanks a lot, can anyone find me any links to questions on CTS? Not ones that involve CTS, but just an equation, or perhaps some of you could give me some? :biggrin:

Thanks

Reply 9

neomilan

17

complete the square:

1.

x^2 - 4x -12 =0

and if you think that's too easy then:

2.

4x^2 + 17x = 6x - 2x^2

Reply 10

Simplicity

17

The Song

That should be banned.

Completing the square is the easiest thing ever. Just think about what happens when you expand something, then you have it.

I think it better to have a understanding of something on a deeper level, you must be doing something wrong.

Completing the square - I don't get it...

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