# How do you factor 3x^2 - 9x + 10?

May 7, 2015

$3 {x}^{2} - 9 x + 10$ has no proper factors.

To see this note that there are no common factors of the 3 terms of this expression (other than $\pm 1$); so it can't be simplified in this way.

Second, consider the discriminant of this expression:
$\Delta = {b}^{2} - 4 a c$
$= 81 - 120$
$< 0$
So $3 {x}^{2} - 9 x + 10 = 0$ has no real solutions
$\rightarrow 3 {x}^{2} - 9 x + 10$ can not be factored into anything of the form:
$\left(a x + b\right) \left(c x + d\right)$