Solve Using the Quadratic Formula 5x^2+5x-1=0

$5 x^{2} + 5 x - 1 = 0$

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 $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $5$ .

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

Multiply $- 20$ by $- 1$ .

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

Add $25$ and $20$ .

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

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

Factor $9$ out of $45$ .

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

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

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

Pull terms out from under the radical.

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

Multiply $2$ by $5$ .

$x = \frac{- 5 \pm 3 \sqrt{5}}{10}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{- 5 \pm 3 \sqrt{5}}{10}$

Decimal Form:

$x = 0.17082039 \dots, - 1.17082039 \dots$