How do you solve 3x^2+2=4x using the quadratic formula?

Feb 8, 2017

The solutions are $S = \left\{\frac{2}{3} + \frac{1}{3} \sqrt{2} i , \frac{2}{3} - \frac{1}{3} \sqrt{2} i\right\}$

Explanation:

We compare this equation to

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

$3 {x}^{2} + 2 = 4 x$

$3 {x}^{2} - 4 x + 2 = 0$

We calculate the discriminant

$\Delta = {b}^{2} - 4 a c$

$\Delta = {\left(- 4\right)}^{2} - 4 \left(3\right) \left(2\right) = 16 - 24 = - 8$

As, $\Delta < 0$, the solutions are not in $\mathbb{R}$ but in $\mathbb{C}$

$x = \frac{- b \pm \sqrt{\Delta}}{2 a}$

$x = \frac{4 \pm \sqrt{- 8}}{6}$

${x}_{1} = \frac{4 + 2 \sqrt{2} i}{6} = \frac{2}{3} + \frac{1}{3} \sqrt{2} i$

${x}_{2} = \frac{4 - 2 \sqrt{2} i}{6} = \frac{2}{3} - \frac{1}{3} \sqrt{2} i$