# The Difference of Two Squares

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## The Difference of Two Squares

The difference of two squares is precisely that. One squared thing subtracted from another squared thing. This means they are straightforward to factorise. Make sure your happy with the following topics before continuing.

Level 6-7GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Difference of Two Squares Formula

The trick to factorising the difference of two squares is to use the formula,

$\textcolor{blue}{a^2 -b^2 = (a+b)(a-b)}$

This can be used in either direction to factorise or expand such expressions quickly. To show that this works, we will expand the two brackets of the general formula.

\begin{aligned} \textcolor{blue}{(a+b)(a-b)} & =\textcolor{black}{ a^2 +\cancel{ab} - \cancel{ab} -b^2 }\\ &= \textcolor{blue}{a^2 -b^2} \end{aligned}

The negative sign arises as you are multiplying a negative by a positive thus the result is negative. As you can see the two ‘cross terms’ involving both $a$ and $b$ cancel, so we are left with just the square terms.

Level 6-7GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE
Level 6-7GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Example 1: Factorising a Simple Quadratic

Factorise $x^2 - 9$.

[2 marks]

Identifying that this is a difference of two squares, $a$ and $b$ are simply the square roots of $x^2$ and $9$ respectively. Hence $a=\sqrt{x^2}=x$ and $b=\sqrt{9}=3$. So the factorisation is,

$(x+3)(x-3)$

Level 6-7GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Example 2: Removing a Common Factor

Factorise $4x^2 - 100y^2$.

[3 marks]

This is still a question involving the difference of two squares however a factor of $4$ has to be taken out first,

$4x^2 - 100y^2 = 4(x^2-25y^2)$

Now we can find $a$ and $b$ by taking the square roots of $x^2$ and $25y^2$ respectively. Hence $a=\sqrt{x^2}=x$ and $b=\sqrt{25y^2}=5y$. So the factorisation is,

$4(x+5y)(x-5y)$

Level 6-7GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## Example 3: Factorising a Quadratic involving Surds

Factorise $y^2 - 7$.

[2 marks]

In some cases the values we find for $a$ or $b$ will be a surd (non-terminating square root) . In this instance $a=\sqrt{y^2}=y$ and $b=\sqrt{7}=\sqrt{7}$. So the factorisation is,

$(y+\sqrt{7})(y-\sqrt{7})$

Level 6-7GCSEAQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

## The Difference of Two Squares Example Questions

Question 1:

Factorise $9x^2 - 49y^2$

[3 marks]

Level 6-7GCSE AQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

Identifying that both of the coefficients of each term are square numbers, then the square roots are $a=\sqrt{9x^2}=3x$ and $b=\sqrt{49y^2}=7y$. So the factorisation is,

$(3x+7y)(3x-7y)$

Gold Standard Education

Question 2:

Factorise $2x^2 - 8$

[3 marks]

Level 6-7GCSE AQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

Here we can remove a factor of $2$ first so,

$2x^2 - 8 =2(x^2 - 4)$

Thus the factorisation is,

$2(x + 2)(x - 2)$

Gold Standard Education

Question 3:

Solve $99^2-98^2$ without using a calculator.

[2 marks]

Level 8-9GCSE AQAEdexcelOCRWJECCambridge iGCSEEdexcel iGCSE

This is a difference of two squares so we can apply the formula the same way,

$99^2-98^2=(99+98)(99-98) = (197)(1)=197$