Question

How do you solve `x² - x - 20 = 0`?

Answer 1

Find two numbers whose product is `20` and whose difference is `1`. The pair `5`, `4` works.

Hence `x^2-x-20 = (x-5)(x+4)`.

So solutions are `x=5` or `x=-4`.

Explanation:

`(x-a)(x+b) = x^2 - (a-b)x - ab`

Matching this against `x^2-x-20` we see that if we can find `a` and `b` such that `a - b = 1` and `ab = 20`, then we can factor the quadratic into two linear terms.

The pair `a=5`, `b=4` works, giving us

`x^2-x-20 = (x-5)(x+4)`

This will be `0` when `(x-5) = 0` or `(x+4) = 0`, which is when `x=5` or `x=-4`.