Find All Complex Solutions -7x^2+12x=10

$- 7 x^{2} + 12 x = 10$

Step 1

Subtract $10$ from both sides of the equation.

$- 7 x^{2} + 12 x - 10 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = - 7$ , $b = 12$ , and $c = - 10$ into the quadratic formula and solve for $x$ .

$\frac{- 12 \pm \sqrt{12^{2} - 4 \cdot (- 7 \cdot - 10)}}{2 \cdot - 7}$

Step 4

Simplify.

Simplify the numerator.

Raise $12$ to the power of $2$ .

$x = \frac{- 12 \pm \sqrt{144 - 4 \cdot - 7 \cdot - 10}}{2 \cdot - 7}$

Multiply $- 4 \cdot - 7 \cdot - 10$ .

Multiply $- 4$ by $- 7$ .

$x = \frac{- 12 \pm \sqrt{144 + 28 \cdot - 10}}{2 \cdot - 7}$

Multiply $28$ by $- 10$ .

$x = \frac{- 12 \pm \sqrt{144 - 280}}{2 \cdot - 7}$

Subtract $280$ from $144$ .

$x = \frac{- 12 \pm \sqrt{- 136}}{2 \cdot - 7}$

Rewrite $- 136$ as $- 1 (136)$ .

$x = \frac{- 12 \pm \sqrt{- 1 \cdot 136}}{2 \cdot - 7}$

Rewrite $\sqrt{- 1 (136)}$ as $\sqrt{- 1} \cdot \sqrt{136}$ .

$x = \frac{- 12 \pm \sqrt{- 1} \cdot \sqrt{136}}{2 \cdot - 7}$

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

$x = \frac{- 12 \pm i \cdot \sqrt{136}}{2 \cdot - 7}$

Rewrite $136$ as $2^{2} \cdot 34$ .

Factor $4$ out of $136$ .

$x = \frac{- 12 \pm i \cdot \sqrt{4 (34)}}{2 \cdot - 7}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 12 \pm i \cdot \sqrt{2^{2} \cdot 34}}{2 \cdot - 7}$

Pull terms out from under the radical.

$x = \frac{- 12 \pm i \cdot (2 \sqrt{34})}{2 \cdot - 7}$

Move $2$ to the left of $i$ .

$x = \frac{- 12 \pm 2 i \sqrt{34}}{2 \cdot - 7}$

Multiply $2$ by $- 7$ .

$x = \frac{- 12 \pm 2 i \sqrt{34}}{- 14}$

Simplify $\frac{- 12 \pm 2 i \sqrt{34}}{- 14}$ .

$x = \frac{6 \pm i \sqrt{34}}{7}$