Solve the following equation by factorisation:

$x^2 - (1 + \sqrt{2})x + \sqrt{2} = 0$

Given,

$x^2 - (1 + \sqrt{2})x + \sqrt{2} = 0 \\[1em] \Rightarrow x^2 - x - \sqrt{2}x + \sqrt{2} = 0 \\[1em] \Rightarrow x(x - 1) - \sqrt{2}(x - 1) = 0 \\[1em] \Rightarrow (x - \sqrt{2})(x - 1) = 0 \text{ (Factorising left side) } \\[1em] x - \sqrt{2} = 0 \text{ or } x - 1 = 0 \text{ (Zero-product rule) } \\[1em] x = \sqrt{2} \text{ or } x = 1$

Hence, the roots of given equation are $\sqrt{2}$ , 1.