Factorise the following:

x² - 8x + 7

To factorise x² - 8x + 7, we want to find two real numbers whose sum is -8 and product is 7. By trial, we see that (-7) + (-1) = -8 and -7 × -1 = 7.

∴ x² - 8x + 7 = x² - 7x + (-1x) + 7

= x² - 7x - x + 7

= x(x - 7) - 1(x - 7)

= (x - 1)(x - 7).

Hence, x² - 8x + 7 = (x - 1)(x - 7).