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

May 18, 2016

$- 7 \left({x}^{2} - 2 x + 3\right)$

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:
$f \left(x\right) = - 7 \left({x}^{2} - 2 x + 3\right)$

NOTE. We can factor f(x) by using the 2 complex roots.
${d}^{2} = - 8$ --> $d = \pm 2 i \sqrt{2}$
$x = - \frac{b}{2 a} \pm \frac{d}{2 a} = \frac{2}{2} \pm \frac{2 i \sqrt{2}}{2} = 1 \pm i \sqrt{2}$
The factored form will be:
$f \left(x\right) = - 7 \left(x - x 1\right) \left(x - x 2\right) = - 7 \left(x - 1 + i \sqrt{2}\right) \left(x - 1 - i \sqrt{2}\right)$