Find All Complex Solutions 3x+2x^2=5x^2-6

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

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

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

Subtract $5 x^{2}$ from both sides of the equation.

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

Add $6$ to both sides of the equation.

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

Subtract $5 x^{2}$ from $2 x^{2}$ .

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

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 3$ , $b = 3$ , and $c = 6$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (- 3 \cdot 6)}}{2 \cdot - 3}$

Simplify.

Simplify the numerator.

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 3 \cdot 6}}{2 \cdot - 3}$

Multiply $- 4$ by $- 3$ .

$x = \frac{- 3 \pm \sqrt{9 + 12 \cdot 6}}{2 \cdot - 3}$

Multiply $12$ by $6$ .

$x = \frac{- 3 \pm \sqrt{9 + 72}}{2 \cdot - 3}$

Add $9$ and $72$ .

$x = \frac{- 3 \pm \sqrt{81}}{2 \cdot - 3}$

Rewrite $81$ as $9^{2}$ .

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

Pull terms out from under the radical, assuming positive real numbers.

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

Multiply $2$ by $- 3$ .

$x = \frac{- 3 \pm 9}{- 6}$

Simplify $\frac{- 3 \pm 9}{- 6}$ .

$x = \frac{1 \pm 3}{2}$

The final answer is the combination of both solutions.

$x = 2, - 1$