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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

\[ \,\,\, 1) \quad x^2 - 1 \] \[ \,\,\, 2) \quad x^2 - 36 \] \[ \,\,\, 3) \quad x^2 - 64 \] \[ \,\,\, 4) \quad x^2 - 16 \] \[ \,\,\, 5) \quad x^2 - 25 \]

\[ \,\,\, 1) \quad x^2 - 1 = (x + 1)(x - 1) \] \[ \,\,\, 2) \quad x^2 - 36 = (x + 6)(x - 6) \] \[ \,\,\, 3) \quad x^2 - 64 = (x + 8)(x - 8) \] \[ \,\,\, 4) \quad x^2 - 16 = (x + 4)(x - 4) \] \[ \,\,\, 5) \quad x^2 - 25 = (x + 5)(x - 5) \]

Exercise #2

\[ \,\,\, 1) \quad x^2 - 66 \] \[ \,\,\, 2) \quad x^2 - 76 \] \[ \,\,\, 3) \quad x^2 - 74 \] \[ \,\,\, 4) \quad x^2 - 99 \] \[ \,\,\, 5) \quad x^2 - 79 \]

\[ \,\,\, 1) \quad x^2 - 66 = (x + \sqrt{66})(x - \sqrt{66}) \] \[ \,\,\, 2) \quad x^2 - 76 = (x + \sqrt{76})(x - \sqrt{76}) \] \[ \,\,\, 3) \quad x^2 - 74 = (x + \sqrt{74})(x - \sqrt{74}) \] \[ \,\,\, 4) \quad x^2 - 99 = (x + \sqrt{99})(x - \sqrt{99}) \] \[ \,\,\, 5) \quad x^2 - 79 = (x + \sqrt{79})(x - \sqrt{79}) \]

Quadratics