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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

\[ \,\,\, 1) \quad x^2 - 100 \] \[ \,\,\, 2) \quad x^2 - 9 \] \[ \,\,\, 3) \quad x^2 - 144 \] \[ \,\,\, 4) \quad x^2 - 1 \] \[ \,\,\, 5) \quad x^2 - 4 \]

\[ \,\,\, 1) \quad x^2 - 100 = (x + 10)(x - 10) \] \[ \,\,\, 2) \quad x^2 - 9 = (x + 3)(x - 3) \] \[ \,\,\, 3) \quad x^2 - 144 = (x + 12)(x - 12) \] \[ \,\,\, 4) \quad x^2 - 1 = (x + 1)(x - 1) \] \[ \,\,\, 5) \quad x^2 - 4 = (x + 2)(x - 2) \]

Exercise #2

\[ \,\,\, 1) \quad x^2 - 13 \] \[ \,\,\, 2) \quad x^2 - 87 \] \[ \,\,\, 3) \quad x^2 - 61 \] \[ \,\,\, 4) \quad x^2 - 12 \] \[ \,\,\, 5) \quad x^2 - 70 \]

\[ \,\,\, 1) \quad x^2 - 13 = (x + \sqrt{13})(x - \sqrt{13}) \] \[ \,\,\, 2) \quad x^2 - 87 = (x + \sqrt{87})(x - \sqrt{87}) \] \[ \,\,\, 3) \quad x^2 - 61 = (x + \sqrt{61})(x - \sqrt{61}) \] \[ \,\,\, 4) \quad x^2 - 12 = (x + \sqrt{12})(x - \sqrt{12}) \] \[ \,\,\, 5) \quad x^2 - 70 = (x + \sqrt{70})(x - \sqrt{70}) \]

Quadratics