Solve Using the Quadratic Formula 5k^2+5k-5=0

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

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

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

Step 3

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $5$ .

$k = \frac{- 5 \pm \sqrt{25 - 20 \cdot - 5}}{2 \cdot 5}$

Multiply $- 20$ by $- 5$ .

$k = \frac{- 5 \pm \sqrt{25 + 100}}{2 \cdot 5}$

Add $25$ and $100$ .

$k = \frac{- 5 \pm \sqrt{125}}{2 \cdot 5}$

Rewrite $125$ as $5^{2} \cdot 5$ .

Factor $25$ out of $125$ .

$k = \frac{- 5 \pm \sqrt{25 (5)}}{2 \cdot 5}$

Rewrite $25$ as $5^{2}$ .

$k = \frac{- 5 \pm \sqrt{5^{2} \cdot 5}}{2 \cdot 5}$

Pull terms out from under the radical.

$k = \frac{- 5 \pm 5 \sqrt{5}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$k = \frac{- 5 \pm 5 \sqrt{5}}{10}$

Simplify $\frac{- 5 \pm 5 \sqrt{5}}{10}$ .

$k = \frac{- 1 \pm \sqrt{5}}{2}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$k = \frac{- 1 \pm \sqrt{5}}{2}$

Decimal Form:

$k = 0.61803398 \dots, - 1.61803398 \dots$