Solve for x 10-9x^2+4x=-6x^2

$10 - 9 x^{2} + 4 x = - 6 x^{2}$

Move all terms containing $x$ to the left side of the equation.

Add $6 x^{2}$ to both sides of the equation.

$10 - 9 x^{2} + 4 x + 6 x^{2} = 0$

Add $- 9 x^{2}$ and $6 x^{2}$ .

$10 - 3 x^{2} + 4 x = 0$

Use the quadratic formula to find the solutions.

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

Substitute the values $a = - 3$ , $b = 4$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{- 4 \pm \sqrt{4^{2} - 4 \cdot (- 3 \cdot 10)}}{2 \cdot - 3}$

Simplify.

Simplify the numerator.

Raise $4$ to the power of $2$ .

$x = \frac{- 4 \pm \sqrt{16 - 4 \cdot - 3 \cdot 10}}{2 \cdot - 3}$

Multiply $- 4$ by $- 3$ .

$x = \frac{- 4 \pm \sqrt{16 + 12 \cdot 10}}{2 \cdot - 3}$

Multiply $12$ by $10$ .

$x = \frac{- 4 \pm \sqrt{16 + 120}}{2 \cdot - 3}$

Add $16$ and $120$ .

$x = \frac{- 4 \pm \sqrt{136}}{2 \cdot - 3}$

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

Factor $4$ out of $136$ .

$x = \frac{- 4 \pm \sqrt{4 (34)}}{2 \cdot - 3}$

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

$x = \frac{- 4 \pm \sqrt{2^{2} \cdot 34}}{2 \cdot - 3}$

Pull terms out from under the radical.

$x = \frac{- 4 \pm 2 \sqrt{34}}{2 \cdot - 3}$

Multiply $2$ by $- 3$ .

$x = \frac{- 4 \pm 2 \sqrt{34}}{- 6}$

Simplify $\frac{- 4 \pm 2 \sqrt{34}}{- 6}$ .

$x = \frac{2 \pm \sqrt{34}}{3}$

The final answer is the combination of both solutions.

$x = \frac{2 + \sqrt{34}}{3}, \frac{2 - \sqrt{34}}{3}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2 + \sqrt{34}}{3}, \frac{2 - \sqrt{34}}{3}$

Decimal Form:

$x = 2.61031729 \dots, - 1.27698396 \dots$