Find All Complex Solutions 2(6x^2+5x)=12

$2 (6 x^{2} + 5 x) = 12$

Step 1

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

Simplify the left side.

Simplify $2 (6 x^{2} + 5 x)$ .

Apply the distributive property.

$2 (6 x^{2}) + 2 (5 x) = 12$

Multiply.

Multiply $6$ by $2$ .

$12 x^{2} + 2 (5 x) = 12$

Multiply $5$ by $2$ .

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

Subtract $12$ from both sides of the equation.

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

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

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

Step 4

Simplify.

Simplify the numerator.

Raise $10$ to the power of $2$ .

$x = \frac{- 10 \pm \sqrt{100 - 4 \cdot 12 \cdot - 12}}{2 \cdot 12}$

Multiply $- 4 \cdot 12 \cdot - 12$ .

Multiply $- 4$ by $12$ .

$x = \frac{- 10 \pm \sqrt{100 - 48 \cdot - 12}}{2 \cdot 12}$

Multiply $- 48$ by $- 12$ .

$x = \frac{- 10 \pm \sqrt{100 + 576}}{2 \cdot 12}$

Add $100$ and $576$ .

$x = \frac{- 10 \pm \sqrt{676}}{2 \cdot 12}$

Rewrite $676$ as $26^{2}$ .

$x = \frac{- 10 \pm \sqrt{26^{2}}}{2 \cdot 12}$

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

$x = \frac{- 10 \pm 26}{2 \cdot 12}$

Multiply $2$ by $12$ .

$x = \frac{- 10 \pm 26}{24}$

Simplify $\frac{- 10 \pm 26}{24}$ .

$x = \frac{- 5 \pm 13}{12}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{2}{3}, - \frac{3}{2}$