Find All Complex Solutions 11x^2-6x-2=0

$11 x^{2} - 6 x - 2 = 0$

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 $a = 11$ , $b = - 6$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{6 \pm \sqrt{{(- 6)}^{2} - 4 \cdot (11 \cdot - 2)}}{2 \cdot 11}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{6 \pm \sqrt{36 - 4 \cdot 11 \cdot - 2}}{2 \cdot 11}$

Multiply $- 4 \cdot 11 \cdot - 2$ .

Multiply $- 4$ by $11$ .

$x = \frac{6 \pm \sqrt{36 - 44 \cdot - 2}}{2 \cdot 11}$

Multiply $- 44$ by $- 2$ .

$x = \frac{6 \pm \sqrt{36 + 88}}{2 \cdot 11}$

Add $36$ and $88$ .

$x = \frac{6 \pm \sqrt{124}}{2 \cdot 11}$

Rewrite $124$ as $2^{2} \cdot 31$ .

Factor $4$ out of $124$ .

$x = \frac{6 \pm \sqrt{4 (31)}}{2 \cdot 11}$

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

$x = \frac{6 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot 11}$

Pull terms out from under the radical.

$x = \frac{6 \pm 2 \sqrt{31}}{2 \cdot 11}$

Multiply $2$ by $11$ .

$x = \frac{6 \pm 2 \sqrt{31}}{22}$

Simplify $\frac{6 \pm 2 \sqrt{31}}{22}$ .

$x = \frac{3 \pm \sqrt{31}}{11}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{3 + \sqrt{31}}{11}, \frac{3 - \sqrt{31}}{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 + \sqrt{31}}{11}, \frac{3 - \sqrt{31}}{11}$

Decimal Form:

$x = 0.77888766 \dots, - 0.23343312 \dots$