Difference of Two Squares

The difference of two squares is a mathematical identity that expresses the difference between two squared terms. It can be represented as :

a^2-b^2=(a+b)(a-b)

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This identity is not in the log tables and must be learnt off.

A square term is a term that can be expressed as a product of the same term :

$9$ can be expressed as $(3)(3)$
$x^2$ can be expressed as $(x)(x)$
$25a^2b^2$ can be expressed as $(5ab)(5ab)$

Example

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Factorise the following expression : $y^2-100$

$y^2$ and $100$ are both square terms separated by a minus (difference), we can apply the identity. It would be helpful to write the expression as $a^2-b^2$ :

(y)^2-(10)^2

Now we can see what $a$ and $b$ are. Let's apply the identity :

(y+10)(y-10)

Example

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Factorise the following expression : $81x^4-16y^8$

Rewrite in the form $a^2-b^2$ :

\left(9x^2\right)^2-\left(4y^4\right)^2

Apply the identity :

\left(9x^2+4y^4\right)\left(9x^2-4y^4\right)

Example

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Factorise the following expression : $2x^3-50x$

In some cases a difference of two squares may only be a sub-step in the factorisation process. First, let's use HCF :

2x(x^2-25)

Observe that within the brackets, we have a difference of two squares :

2x(x+5)(x-5)

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