Find All Complex Solutions x^2+10x-25=0

$x^{2} + 10 x - 25 = 0$

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 $a = 1$ , $b = 10$ , and $c = - 25$ into the quadratic formula and solve for $x$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (1 \cdot - 25)}}{2 \cdot 1}$

Step 3

Simplify.

Simplify the numerator.

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 1 \cdot - 25}}{2 \cdot 1}$

Multiply $- 4 \cdot 1 \cdot - 25$ .

Multiply $- 4$ by $1$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot - 25}}{2 \cdot 1}$

Multiply $- 4$ by $- 25$ .

$x = \frac{- 10 \pm \sqrt{100 + 100}}{2 \cdot 1}$

Add $100$ and $100$ .

$x = \frac{- 10 \pm \sqrt{200}}{2 \cdot 1}$

Rewrite $200$ as $10^{2} \cdot 2$ .

Factor $100$ out of $200$ .

$x = \frac{- 10 \pm \sqrt{100 (2)}}{2 \cdot 1}$

Rewrite $100$ as $10^{2}$ .

$x = \frac{- 10 \pm \sqrt{10^{2} \cdot 2}}{2 \cdot 1}$

Pull terms out from under the radical.

$x = \frac{- 10 \pm 10 \sqrt{2}}{2 \cdot 1}$

Multiply $2$ by $1$ .

$x = \frac{- 10 \pm 10 \sqrt{2}}{2}$

Simplify $\frac{- 10 \pm 10 \sqrt{2}}{2}$ .

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

Step 4

The final answer is the combination of both solutions.

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.07106781 \dots, - 12.07106781 \dots$