Completing the square

Hey everyone 

here are some notes on how to complete the square

Hope they help

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Simple question

Question: Make the square of this expression x^2 - 14x

Method:

Firstly half the number 14/2=7

so our answer is (x - 7)^2   this is how you would write it out, this part is called making the square

Next, factorise the equation (x-7)(x-7) = x^2 - 14x + 49

But this does not equal x^2 - 14x like we needed it to

so the last part in completing the square is the - the 49

So our answer is (x-7)^2 -49

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Medium question

Complete the square of this equation;

x^2 - 14x - 5

We already know that to complete the square of the expression x^2 - 14x from the previous page our answer was ( x - 7 )^2 -49

Now all we need to do is minus the 5 from the -49 of our equation, so  ( x - 7 )^2 - 49 -5 = 

( x - 7 ) ^2 - 54 this is our answer.

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Using completing the square to solve a quadratic e

Question: x^2 + 12x - 5=0

Complete the square first: ( x + 6 )^2 - 41 = 0

Now to solve the equation we add 41 to both sides: ( x + 6 )^2 = 41

We now need to square root each side:  x + 6  = Sqrt41

Now we take away 6 from both sides: x = -6+-sqrt41  (the answer could be positive or negative)

Write our answer in full: - 6 +- 6.4

Our answer is 0.4 (because we do - 6 + 6.4 = 0.4 )

Our answer is also -12.4 (because we do - 6 - 6.4 = -12.4)

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Harder quadratic equations

For example if we have 2x^2 + 4x -22

We cannot complete the square because to complete the square you have to have only 1x^2

So all we have to do is divide the whole equation by 2 so it would become x^2 + 2x - 11

It works the same for any number, say if we have 8x^2, all we do is divide the whole equation by 8 and carry on working out the equation as normal.

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