Difference of Two Squares

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Expanding the expression $(a + 3)(a - 3)$ gives an answer of $a^2 - 9$ : there is no $a$ term.

$a^2 - 9$ can be written as $a^2 - 3^2$ : both terms are squared and the subtraction gives the difference.

Note that the constant in the expanded expression is a negative.

This is known as the difference of two squares.

Example 1

Factorise $b^2 - 81$

No $b$ term and √81 = 9 and -9, so is the difference of two squares.

Answer: $(b + 9)(b - 9)$

Show $a^2x^2 - y^2$ as the difference of two squares.

First term: $sqrt(a^2x^2) = ax$

Second term: $sqrt(y^2) = +y text( and ) -y$

$a^2x^2 - y^2 = (ax + y)(ax - y)$

Answer: $(ax + y)(ax - y)$