Question

How do you factor completely `−7x^2 + 14x − 21`?

Answer 1

`-7(x^2 - 2x + 3)`

Explanation:

f(x) = -7y = -7(x^2 - 2x + 3)
D = d^2 = b^2 - 4ac = 4 - 12 = -8.
Since D < 0, the trinomial in parentheses can't be factored.
Therefor:
`f(x) = -7(x^2 - 2x + 3)`

NOTE. We can factor f(x) by using the 2 complex roots.
`d^2 = -8` --> `d = +- 2isqrt2`
`x = -b/(2a) +- d/(2a) = 2/2 +- (2isqrt2)/2 = 1 +- isqrt2`
The factored form will be:
`f(x) = -7(x - x1)(x - x2) = - 7(x - 1 + isqrt2)(x -1 - isqrt2)`