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
$$x+y$$$$x-y$$
$$x+3$$$$x-3$$
$$x+\sqrt{5}$$$$x-\sqrt{5}$$

#### Factorisation of a difference of two squares (DOTS)

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

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

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