Find All Complex Solutions x^2+4x+2=0

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 1$ , $b = 4$ , and $c = 2$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot 2}}{2 \cdot 1}$

Multiply $- 4$ by $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 8}}{2 \cdot 1}$

Subtract $8$ from $16$ .

$x = \frac{- 4 \pm \sqrt{8}}{2 \cdot 1}$

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $1$ .

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

Simplify $\frac{- 4 \pm 2 \sqrt{2}}{2}$ .

$x = - 2 \pm \sqrt{2}$

The final answer is the combination of both solutions.

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

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = - 0.58578643 \dots, - 3.41421356 \dots$