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

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

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 $a = 2$ , $b = 3$ , and $c = 5$ into the quadratic formula and solve for $x$ .

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

Step 3

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot 2 \cdot 5$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot 5}}{2 \cdot 2}$

Multiply $- 8$ by $5$ .

$x = \frac{- 3 \pm \sqrt{9 - 40}}{2 \cdot 2}$

Subtract $40$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 31}}{2 \cdot 2}$

Rewrite $- 31$ as $- 1 (31)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 31}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (31)}$ as $\sqrt{- 1} \cdot \sqrt{31}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot 2}$

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

$x = \frac{- 3 \pm i \sqrt{31}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm i \sqrt{31}}{4}$