Question

How do you factor `6x^3-6x^2+18x`?

Answer 1

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

Explanation:

Given
`color(white)("XXX")6x^3-6x^2+18x`

Extract the obvious common factor `6x` from each term:
`color(white)("XXX")6x(x^2-x+3)`

We could try to factor `(x^2-x+3)`
but an examination of the discriminant (see below) gives us
`color(white)("XXX")`Delta = (-1)^2-4(1)(3) < 0#
which tells us there are no Real roots;
so factors are not available.

Discriminants and their significance
`color(white)("XXX")`for a quadratic in the form
`color(white)("XXXXX")ax^2+bx+c`
`color(white)("XXX")`the discriminant is
`color(white)("XXXXX")Delta = b^2-4ac`

#color(white)("XXX")Delta { (< 0 rarr "no Real roots"), (= 0 rarr "exactly one Real root"), (> 0 rarr "two Real roots"):}#