Find All Complex Solutions -7x^2+5x-2=0

$- 7 x^{2} + 5 x - 2 = 0$

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 $a = - 7$ , $b = 5$ , and $c = - 2$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (- 7 \cdot - 2)}}{2 \cdot - 7}$

Step 3

Simplify.

Simplify the numerator.

Raise $5$ to the power of $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot - 7 \cdot - 2}}{2 \cdot - 7}$

Multiply $- 4 \cdot - 7 \cdot - 2$ .

Multiply $- 4$ by $- 7$ .

$x = \frac{- 5 \pm \sqrt{25 + 28 \cdot - 2}}{2 \cdot - 7}$

Multiply $28$ by $- 2$ .

$x = \frac{- 5 \pm \sqrt{25 - 56}}{2 \cdot - 7}$

Subtract $56$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 31}}{2 \cdot - 7}$

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

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 31}}{2 \cdot - 7}$

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

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{31}}{2 \cdot - 7}$

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

$x = \frac{- 5 \pm i \sqrt{31}}{2 \cdot - 7}$

Multiply $2$ by $- 7$ .

$x = \frac{- 5 \pm i \sqrt{31}}{- 14}$

Simplify $\frac{- 5 \pm i \sqrt{31}}{- 14}$ .

$x = \frac{5 \pm i \sqrt{31}}{14}$