Solve for x 0=2x^2+3x+4

$0 = 2 x^{2} + 3 x + 4$

Rewrite the equation as $2 x^{2} + 3 x + 4 = 0$ .

$2 x^{2} + 3 x + 4 = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 2$ , $b = 3$ , and $c = 4$ into the quadratic formula and solve for $x$ .

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

Simplify.

Simplify the numerator.

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

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

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot 4}}{2 \cdot 2}$

Multiply $- 8$ by $4$ .

$x = \frac{- 3 \pm \sqrt{9 - 32}}{2 \cdot 2}$

Subtract $32$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 23}}{2 \cdot 2}$

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 2}$

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

$x = \frac{- 3 \pm i \sqrt{23}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm i \sqrt{23}}{4}$

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{23}}{4}, - \frac{3 + i \sqrt{23}}{4}$