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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

\[ \,\,\, 1) \quad x^2 - 121 \] \[ \,\,\, 2) \quad x^2 - 49 \] \[ \,\,\, 3) \quad x^2 - 49 \] \[ \,\,\, 4) \quad x^2 - 36 \] \[ \,\,\, 5) \quad x^2 - 1 \]

\[ \,\,\, 1) \quad x^2 - 121 = (x + 11)(x - 11) \] \[ \,\,\, 2) \quad x^2 - 49 = (x + 7)(x - 7) \] \[ \,\,\, 3) \quad x^2 - 49 = (x + 7)(x - 7) \] \[ \,\,\, 4) \quad x^2 - 36 = (x + 6)(x - 6) \] \[ \,\,\, 5) \quad x^2 - 1 = (x + 1)(x - 1) \]

Exercise #2

\[ \,\,\, 1) \quad x^2 - 32 \] \[ \,\,\, 2) \quad x^2 - 22 \] \[ \,\,\, 3) \quad x^2 - 41 \] \[ \,\,\, 4) \quad x^2 - 94 \] \[ \,\,\, 5) \quad x^2 - 7 \]

\[ \,\,\, 1) \quad x^2 - 32 = (x + \sqrt{32})(x - \sqrt{32}) \] \[ \,\,\, 2) \quad x^2 - 22 = (x + \sqrt{22})(x - \sqrt{22}) \] \[ \,\,\, 3) \quad x^2 - 41 = (x + \sqrt{41})(x - \sqrt{41}) \] \[ \,\,\, 4) \quad x^2 - 94 = (x + \sqrt{94})(x - \sqrt{94}) \] \[ \,\,\, 5) \quad x^2 - 7 = (x + \sqrt{7})(x - \sqrt{7}) \]

Quadratics