Difference of Two Squares
Difference of Two Squares
We have seen that \((ax + b)(ax - b)\) can be expanded to \({a}^{2}{x}^{2} - {b}^{2}\).
Therefore \({a}^{2}{x}^{2} - {b}^{2}\) can be factorised as \((ax + b)(ax - b)\).
For example, \({x}^{2} - 16\) can be written as \({x}^{2} - {4}^{2}\) which is a difference of two squares. Therefore, the factors of \({x}^{2} - 16\) are \((x - 4)\) and \((x + 4)\).
To spot a difference of two squares, look for expressions:
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consisting of two terms;
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with terms that have different signs (one positive, one negative);
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with each term a perfect square.
For example: \({a}^{2} - 1\); \(4{x}^{2} - {y}^{2}\); \(-49 + {p}^{4}\).
The following video explains factorising the difference of two squares.
Example
Question
Factorise: \(3a({a}^{2} - 4) - 7({a}^{2} - 4)\).
Take out the common factor \(({a}^{2} - 4)\)
\[3a({a}^{2} - 4) - 7({a}^{2} - 4) = ({a}^{2} - 4)(3a - 7)\]Factorise the difference of two squares \(({a}^{2} - 4)\)
\[({a}^{2} - 4)(3a - 7) = (a - 2)(a + 2)(3a - 7)\]
This lesson is part of:
Algebraic Expressions Overview