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

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

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

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

Step 3

Simplify.

Simplify the numerator.

One to any power is one.

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

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

Multiply $- 4$ by $5$ .

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

Multiply $- 20$ by $1$ .

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

Subtract $20$ from $1$ .

$x = \frac{- 1 \pm \sqrt{- 19}}{2 \cdot 5}$

Rewrite $- 19$ as $- 1 (19)$ .

$x = \frac{- 1 \pm \sqrt{- 1 \cdot 19}}{2 \cdot 5}$

Rewrite $\sqrt{- 1 (19)}$ as $\sqrt{- 1} \cdot \sqrt{19}$ .

$x = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{19}}{2 \cdot 5}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 1 \pm i \sqrt{19}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{- 1 \pm i \sqrt{19}}{10}$