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

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

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 $a = 1$ , $b = 6$ , and $c = 2$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot 1 \cdot 2$ .

Multiply $- 4$ by $1$ .

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

Multiply $- 4$ by $2$ .

$x = \frac{- 6 \pm \sqrt{36 - 8}}{2 \cdot 1}$

Subtract $8$ from $36$ .

$x = \frac{- 6 \pm \sqrt{28}}{2 \cdot 1}$

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$x = \frac{- 6 \pm \sqrt{4 (7)}}{2 \cdot 1}$

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

$x = \frac{- 6 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 6 \pm 2 \sqrt{7}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 6 \pm 2 \sqrt{7}}{2}$

Simplify $\frac{- 6 \pm 2 \sqrt{7}}{2}$ .

$x = - 3 \pm \sqrt{7}$

Step 4

The final answer is the combination of both solutions.

$x = - 3 + \sqrt{7}, - 3 - \sqrt{7}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - 3 + \sqrt{7}, - 3 - \sqrt{7}$

Decimal Form:

$x = - 0.35424868 \dots, - 5.64575131 \dots$