Solve Using the Quadratic Formula 3x-2x^2=7

$3 x - 2 x^{2} = 7$

Step 1

Subtract $7$ from both sides of the equation.

$3 x - 2 x^{2} - 7 = 0$

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 $a = - 2$ , $b = 3$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

Simplify the numerator.

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

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

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

Multiply $- 4$ by $- 2$ .

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

Multiply $8$ by $- 7$ .

$x = \frac{- 3 \pm \sqrt{9 - 56}}{2 \cdot - 2}$

Subtract $56$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 47}}{2 \cdot - 2}$

Rewrite $- 47$ as $- 1 (47)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 47}}{2 \cdot - 2}$

Rewrite $\sqrt{- 1 (47)}$ as $\sqrt{- 1} \cdot \sqrt{47}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{47}}{2 \cdot - 2}$

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

$x = \frac{- 3 \pm i \sqrt{47}}{2 \cdot - 2}$

Multiply $2$ by $- 2$ .

$x = \frac{- 3 \pm i \sqrt{47}}{- 4}$

Simplify $\frac{- 3 \pm i \sqrt{47}}{- 4}$ .

$x = \frac{3 \pm i \sqrt{47}}{4}$