How do you solve 3x^2 - 12x + 2 = 0?

May 10, 2017

$x = 2 \pm \frac{\sqrt{30}}{3}$

Explanation:

The equation:

$3 {x}^{2} - 12 x + 2 = 0$

is in the standard form:

$a {x}^{2} + b x + c = 0$

with $a = 3$, $b = - 12$ and $c = 2$

The discriminant $\Delta$ of a quadratic is given by the formula:

$\Delta = {b}^{2} - 4 a c = {\left(\textcolor{b l u e}{- 12}\right)}^{2} - 4 \left(\textcolor{b l u e}{3}\right) \left(\textcolor{b l u e}{2}\right) = 144 - 24 = 120 = {2}^{2} \cdot 30$

Since $\Delta > 0$, the given quadratic equation has two Real roots, but since $\Delta$ is not a perfect square those roots are irrational.

We can find the roots using the quadratic formula:

$x = \frac{- b \pm \sqrt{{b}^{2} - 4 a c}}{2 a}$

$\textcolor{w h i t e}{x} = \frac{- b \pm \sqrt{\Delta}}{2 a}$

$\textcolor{w h i t e}{x} = \frac{12 \pm \sqrt{120}}{6}$

$\textcolor{w h i t e}{x} = \frac{12 \pm \sqrt{{2}^{2} \cdot 30}}{6}$

$\textcolor{w h i t e}{x} = \frac{12 \pm 2 \sqrt{30}}{6}$

$\textcolor{w h i t e}{x} = 2 \pm \frac{\sqrt{30}}{3}$

That is:

$x = 2 + \frac{\sqrt{30}}{3} \text{ }$ or $\text{ } x = 2 - \frac{\sqrt{30}}{3}$