How do I complete the square of an equation?

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First off, write your equation in the form:

ax^2+by+c=0

with a, b and c being your constant coefficients. 

First, we will factorise out a such that:

a(x^+(b/a)x)+c

Now divide b by 2. Then write 

a(x+(b/2a))^2

If you multiply this out you should get a(x^+(b/a)x+d)

Then times everything by a again to give ax^2+bx+ad

The final stage is to add a constant on the end which will get you from ad to c in the final expansion.

To do this we want ad+m=c where m is the constant we are trying to find. Therefore m=c-ad.

Our final answer will be:

ax^+bx+c=a(x+(b/2a))^+m .

 

This is much easier to see with an example:

Complete the square of 4x^2+4x=-6

We need to rearrange this to 4x^2+4x+6=0.

First we factorise out the 4:

4(x^2+x)+6=0 

Now we follow through the rest of the steps:

4(x+1/2)^2=4(x^2+x+1/4)=4x^2+4x+1

1+m=6

m=5

So the final answer is

4x^2+4x+6=4(x+1/2)^2+5=0

Beth D. GCSE Maths tutor, A Level Maths tutor, GCSE Music tutor

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