Question

How do you factor the expression `x^2-x-36`?

Answer 1

Use the quadratic formula to find:

`x^2-x-36 = (x-(1-sqrt(145))/2)(x-(1+sqrt(145))/2)`

Explanation:

You would like to find a pair of factors of `36` that differ by `1`. Unfortunately, no such pair exists, so this polynomial will not factor with integer coefficients.

Let's check the discriminant: `x^2-x-36` is in the form `ax+bx+c` with `a=1`, `b=-1` and `c=-36`. This has discriminant `Delta` given by the formula:

`Delta = b^2-4ac = (-1)^2-(4*1*-36) = 1+144 = 145`

Since this is positive but not a perfect square, the quadratic has factors with Real irrational coefficients.

We can find these factors using the quadratic formula. The zeros of the quadratic are given by:

`x = (-b+-sqrt(b^2-4ac))/(2a) = (-b+-sqrt(Delta))/(2a)`

`=(1+-sqrt(145))/2`

That is `x_1 = (1-sqrt(145))/2` and `x_2 = (1+sqrt(145))/2`

So:

`x^2-x-36 = (x-x_1)(x-x_2) = (x-(1-sqrt(145))/2)(x-(1+sqrt(145))/2)`