Question

How do you factor the expression `x^2 - x - 6`?

Answer 1

`x^2 - x - 6 = (x - 3)(x + 2)`

Explanation:

There are many possible ways to gain the factorisation. Let me show you one of them.

You would like to have something like

`x ^2 - x - 6 = (x + a)(x + b)`
`color(white)(xxxxxxxii) = x^2 + (a+b)x + a*b`

So, your goal is to determine `a` and `b` so that

`{ (a + b = -1), ( color(white)(x) a * b = -6) :}`

hold.

This approach is not always possible or can be complicated if your solutions are not integers.
Here, however, it's one of the easiest ways to compute the result. :-)

You know that `a * b = -6` can be achieved either with `-3 * 2 = -6` or with `3 * (-2) = -6`.

Let's test which pair works for the first equation:
`-3 + 2 = -1` works perfectly well, so it's `a = -3` and `b = 2`.

Thus, your factorization is:

`x^2 - x - 6 = (x - 3)(x + 2)`.