The Difference of Two Squares

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.

• Factorising Quadratics

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,

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

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,

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

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,

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