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Difference of two squares (DOTS)
\[ a^2 - b^2 = (a + b)(a - b) \]
The expressions \(a + b\) and \(a - b\) are called conjugates.

Conjugates are formed by changing the sign between two terms.

More examples:

ExpressionConjugate
\(x+y\)\(x-y\)
\(x+3\)\(x-3\)
\(x+\sqrt{5}\)\(x-\sqrt{5}\)

Factorise the following expressions:

Exercise #1

\[ \,\,\, 1) \quad x^2 - 64 \] \[ \,\,\, 2) \quad x^2 - 4 \] \[ \,\,\, 3) \quad x^2 - 4 \] \[ \,\,\, 4) \quad x^2 - 49 \] \[ \,\,\, 5) \quad x^2 - 9 \]

\[ \,\,\, 1) \quad x^2 - 64 = (x + 8)(x - 8) \] \[ \,\,\, 2) \quad x^2 - 4 = (x + 2)(x - 2) \] \[ \,\,\, 3) \quad x^2 - 4 = (x + 2)(x - 2) \] \[ \,\,\, 4) \quad x^2 - 49 = (x + 7)(x - 7) \] \[ \,\,\, 5) \quad x^2 - 9 = (x + 3)(x - 3) \]

Exercise #2

\[ \,\,\, 1) \quad x^2 - 2 \] \[ \,\,\, 2) \quad x^2 - 89 \] \[ \,\,\, 3) \quad x^2 - 81 \] \[ \,\,\, 4) \quad x^2 - 22 \] \[ \,\,\, 5) \quad x^2 - 40 \]

\[ \,\,\, 1) \quad x^2 - 2 = (x + \sqrt{2})(x - \sqrt{2}) \] \[ \,\,\, 2) \quad x^2 - 89 = (x + \sqrt{89})(x - \sqrt{89}) \] \[ \,\,\, 3) \quad x^2 - 81 = (x + 9)(x - 9) \] \[ \,\,\, 4) \quad x^2 - 22 = (x + \sqrt{22})(x - \sqrt{22}) \] \[ \,\,\, 5) \quad x^2 - 40 = (x + \sqrt{40})(x - \sqrt{40}) \]

Quadratics