Find All Complex Solutions 6x^2-6x-2=0

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

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 6$ , $b = - 6$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot 6 \cdot - 2$ .

Multiply $- 4$ by $6$ .

$x = \frac{6 \pm \sqrt{36 - 24 \cdot - 2}}{2 \cdot 6}$

Multiply $- 24$ by $- 2$ .

$x = \frac{6 \pm \sqrt{36 + 48}}{2 \cdot 6}$

Add $36$ and $48$ .

$x = \frac{6 \pm \sqrt{84}}{2 \cdot 6}$

Rewrite $84$ as $2^{2} \cdot 21$ .

Factor $4$ out of $84$ .

$x = \frac{6 \pm \sqrt{4 (21)}}{2 \cdot 6}$

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 21}}{2 \cdot 6}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{21}}{2 \cdot 6}$

Multiply $2$ by $6$ .

$x = \frac{6 \pm 2 \sqrt{21}}{12}$

Simplify $\frac{6 \pm 2 \sqrt{21}}{12}$ .

$x = \frac{3 \pm \sqrt{21}}{6}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{21}}{6}, \frac{3 - \sqrt{21}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{21}}{6}, \frac{3 - \sqrt{21}}{6}$

Decimal Form:

$x = 1.26376261 \dots, - 0.26376261 \dots$