Question

How do you factor `x^2 -x+7`?

Answer 1

Use the quadratic formula to find:

`x^2-x+7 = (x-1/2-(3sqrt(3))/2 i)(x-1/2+(3sqrt(3))/2 i)`

Explanation:

`f(x) = x^2-x+7` is in the form `ax^2+bx+c` with `a=1`, `b=-1` and `c=7`

This has discriminant `Delta` given by the formula:

`Delta = b^2-4ac = (-1)^2-(4*1*7) = 1-28 = -27`

Since this is negative `f(x)` has no linear factors with Real coefficients. We can find its Complex factorisation by using the quadratic formula then converting the zeros into factors:

The roots of `f(x) = 0` are given by the quadratic formula:

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

`= (-b+-sqrt(Delta))/(2a)`

`= (1+-sqrt(-27))/2`

`= 1/2 +- sqrt(27)/2 i`

`= 1/2 +- (3sqrt(3))/2 i`

Hence `f(x)` can be factored as:

`x^2-x+7 = (x-1/2-(3sqrt(3))/2 i)(x-1/2+(3sqrt(3))/2 i)`