Find All Complex Solutions 7x^2+3x+8=0

$7 x^{2} + 3 x + 8 = 0$

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 $a = 7$ , $b = 3$ , and $c = 8$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (7 \cdot 8)}}{2 \cdot 7}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 7 \cdot 8}}{2 \cdot 7}$

Multiply $- 4 \cdot 7 \cdot 8$ .

Multiply $- 4$ by $7$ .

$x = \frac{- 3 \pm \sqrt{9 - 28 \cdot 8}}{2 \cdot 7}$

Multiply $- 28$ by $8$ .

$x = \frac{- 3 \pm \sqrt{9 - 224}}{2 \cdot 7}$

Subtract $224$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 215}}{2 \cdot 7}$

Rewrite $- 215$ as $- 1 (215)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 215}}{2 \cdot 7}$

Rewrite $\sqrt{- 1 (215)}$ as $\sqrt{- 1} \cdot \sqrt{215}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{215}}{2 \cdot 7}$

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

$x = \frac{- 3 \pm i \sqrt{215}}{2 \cdot 7}$

Multiply $2$ by $7$ .

$x = \frac{- 3 \pm i \sqrt{215}}{14}$