Solve Using the Quadratic Formula 5x^2+3x+1=0

$5 x^{2} + 3 x + 1 = 0$

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 $a = 5$ , $b = 3$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (5 \cdot 1)}}{2 \cdot 5}$

Step 3

Simplify.

Simplify the numerator.

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

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

Multiply $- 4$ by $5$ .

$x = \frac{- 3 \pm \sqrt{9 - 20 \cdot 1}}{2 \cdot 5}$

Multiply $- 20$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 20}}{2 \cdot 5}$

Subtract $20$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 11}}{2 \cdot 5}$

Rewrite $- 11$ as $- 1 (11)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 11}}{2 \cdot 5}$

Rewrite $\sqrt{- 1 (11)}$ as $\sqrt{- 1} \cdot \sqrt{11}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{11}}{2 \cdot 5}$

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

$x = \frac{- 3 \pm i \sqrt{11}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{- 3 \pm i \sqrt{11}}{10}$