Find All Complex Solutions 3x^2-1=6x

$3 x^{2} - 1 = 6 x$

Step 1

Subtract $6 x$ from both sides of the equation.

$3 x^{2} - 1 - 6 x = 0$

Step 2

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 3

Substitute the values $a = 3$ , $b = - 6$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (3 \cdot - 1)}}{2 \cdot 3}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 6$ to the power of $2$ .

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 3 \cdot - 1}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot - 1$ .

Multiply $- 4$ by $3$ .

$x = \frac{6 \pm \sqrt{36 - 12 \cdot - 1}}{2 \cdot 3}$

Multiply $- 12$ by $- 1$ .

$x = \frac{6 \pm \sqrt{36 + 12}}{2 \cdot 3}$

Add $36$ and $12$ .

$x = \frac{6 \pm \sqrt{48}}{2 \cdot 3}$

Rewrite $48$ as $4^{2} \cdot 3$ .

Factor $16$ out of $48$ .

$x = \frac{6 \pm \sqrt{16 (3)}}{2 \cdot 3}$

Rewrite $16$ as $4^{2}$ .

$x = \frac{6 \pm \sqrt{4^{2} \cdot 3}}{2 \cdot 3}$

Pull terms out from under the radical.

$x = \frac{6 \pm 4 \sqrt{3}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{6 \pm 4 \sqrt{3}}{6}$

Simplify $\frac{6 \pm 4 \sqrt{3}}{6}$ .

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

Step 5

The final answer is the combination of both solutions.

$x = \frac{3 + 2 \sqrt{3}}{3}, \frac{3 - 2 \sqrt{3}}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + 2 \sqrt{3}}{3}, \frac{3 - 2 \sqrt{3}}{3}$

Decimal Form:

$x = 2.15470053 \dots, - 0.15470053 \dots$