The Difference of 2 squares

This is a continuation of Quadratic Equations
This question is very common in GCSE papers and there is no trick to it you must just learn to recognise it

You may get a question asking

Factorise x^2 - 16
This is the difference of two squares, as you are subtracting (finding the difference between) two square numbers

You must notice that a square number cannot be negative but those in The Difference of 2 squares are.
Now this is still recognisable as a quadratic as it could be rewritten as
x^2 +/-0x - 16

This means the two numbers multiply to make 16 and add to make 0.
and as this is a square number, those numbers must be 4's
and since the 16 is negative one of the 4 must be negative and one must be positive
as 2 negatives make a positive.

So we will write it out as
(x-4)(x+4)

which multiplies out as

x times x = x^2
x times -4 = -4x
x times 4 = 4x
-4 times 4 = -16
Then x^2 -4x+4x-16
And if we tidy it up
x^2 -16 then we know we have the write answer
and if the question asks
x^2 -16=0
then just follow the example in the Quadratic Equations section

Factorising a Quadratic

For this revision lesson we will look at factorising a quadratic.
(x^2=x "squared")
for example

x^2 +2x -15

so to factorise we need to "whack it into brackets"

so to make the x^2 we need 2 sets of brackets so write it out like

( )( )

then how do we make the x^2?

(x )(x ) there now we have x multiplied by x so it makes x^2

Now we must find two numbers that multiply to make negative 15 and add together to make 2
so we shall list out the possibilities for the multiplication

-1 & 15
-15 & 1
-3 & 5
-5 & 3

Now which pair add to make positive 2
-5+3 = -2
-1+15=14
-15+1=-14
-3+5=2

Now we have our solution we put it into the brackets

(x-3)(x+5) and then multiply out to check

So:
x times x = x^2
x times 5 = +5x
x times - 3 = -3x
3 times -5 = -15

so x^2 +5x -3x - 15

tidy it up : x^2 +2x-15 and there we go it matches up and is correct

Now some questions will go on to ask

x^2 + 2x -15 = 0

Now if anything multiplies to make zero then one of the digits must be zero

so if (x-3)(x+5) =x^2 + 2x -15

and x^2 + 2x -15 = 0

then (x-3)(x+5) =x^2 + 2x -15 =0

so (x-3)(x+5) =0
so if one of the brackets has to equal zero then
(x-3)= 0
so we take the 3 over to the other side and it becomes positive
so x=3
but the other bracket could also equal zero so
(x+5)=0
so x=-5

that means x= 3 or -5