Find All Complex Solutions x^2+5x-5=0

$x^{2} + 5 x - 5 = 0$

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 $a = 1$ , $b = 5$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot - 5)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 5$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 5}}{2 \cdot 1}$

Multiply $- 4$ by $- 5$ .

$x = \frac{- 5 \pm \sqrt{25 + 20}}{2 \cdot 1}$

Add $25$ and $20$ .

$x = \frac{- 5 \pm \sqrt{45}}{2 \cdot 1}$

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$x = \frac{- 5 \pm \sqrt{9 (5)}}{2 \cdot 1}$

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 5 \pm \sqrt{3^{2} \cdot 5}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 5 \pm 3 \sqrt{5}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm 3 \sqrt{5}}{2}$

Step 4

The final answer is the combination of both solutions.

$x = - \frac{5 - 3 \sqrt{5}}{2}, - \frac{5 + 3 \sqrt{5}}{2}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = - \frac{5 - 3 \sqrt{5}}{2}, - \frac{5 + 3 \sqrt{5}}{2}$

Decimal Form:

$x = 0.85410196 \dots, - 5.85410196 \dots$