Solve Using the Quadratic Formula 3x+5x^2=x-6

$3 x + 5 x^{2} = x - 6$

Move all terms to the left side of the equation and simplify.

Move all the expressions to the left side of the equation.

Subtract $x$ from both sides of the equation.

$3 x + 5 x^{2} - x = - 6$

Add $6$ to both sides of the equation.

$3 x + 5 x^{2} - x + 6 = 0$

Subtract $x$ from $3 x$ .

$5 x^{2} + 2 x + 6 = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = 5$ , $b = 2$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (5 \cdot 6)}}{2 \cdot 5}$

Simplify.

Simplify the numerator.

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 5 \cdot 6}}{2 \cdot 5}$

Multiply $- 4$ by $5$ .

$x = \frac{- 2 \pm \sqrt{4 - 20 \cdot 6}}{2 \cdot 5}$

Multiply $- 20$ by $6$ .

$x = \frac{- 2 \pm \sqrt{4 - 120}}{2 \cdot 5}$

Subtract $120$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 116}}{2 \cdot 5}$

Rewrite $- 116$ as $- 1 (116)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 116}}{2 \cdot 5}$

Rewrite $\sqrt{- 1 (116)}$ as $\sqrt{- 1} \cdot \sqrt{116}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{116}}{2 \cdot 5}$

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

$x = \frac{- 2 \pm i \cdot \sqrt{116}}{2 \cdot 5}$

Rewrite $116$ as $2^{2} \cdot 29$ .

Factor $4$ out of $116$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (29)}}{2 \cdot 5}$

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 29}}{2 \cdot 5}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{29})}{2 \cdot 5}$

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

$x = \frac{- 2 \pm 2 i \sqrt{29}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{- 2 \pm 2 i \sqrt{29}}{10}$

Simplify $\frac{- 2 \pm 2 i \sqrt{29}}{10}$ .

$x = \frac{- 1 \pm i \sqrt{29}}{5}$