Solve for y 2+(3y-2)y=-6

$2 + (3 y - 2) y = - 6$

Step 1

Simplify each term.

Apply the distributive property.

$2 + 3 y \cdot y - 2 y = - 6$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$2 + 3 (y \cdot y) - 2 y = - 6$

Multiply $y$ by $y$ .

$2 + 3 y^{2} - 2 y = - 6$

Step 2

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

Add $6$ to both sides of the equation.

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

Add $2$ and $6$ .

$3 y^{2} - 2 y + 8 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 3$ , $b = - 2$ , and $c = 8$ into the quadratic formula and solve for $y$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (3 \cdot 8)}}{2 \cdot 3}$

Step 5

Simplify.

Simplify the numerator.

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

$y = \frac{2 \pm \sqrt{4 - 4 \cdot 3 \cdot 8}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 8$ .

Multiply $- 4$ by $3$ .

$y = \frac{2 \pm \sqrt{4 - 12 \cdot 8}}{2 \cdot 3}$

Multiply $- 12$ by $8$ .

$y = \frac{2 \pm \sqrt{4 - 96}}{2 \cdot 3}$

Subtract $96$ from $4$ .

$y = \frac{2 \pm \sqrt{- 92}}{2 \cdot 3}$

Rewrite $- 92$ as $- 1 (92)$ .

$y = \frac{2 \pm \sqrt{- 1 \cdot 92}}{2 \cdot 3}$

Rewrite $\sqrt{- 1 (92)}$ as $\sqrt{- 1} \cdot \sqrt{92}$ .

$y = \frac{2 \pm \sqrt{- 1} \cdot \sqrt{92}}{2 \cdot 3}$

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

$y = \frac{2 \pm i \cdot \sqrt{92}}{2 \cdot 3}$

Rewrite $92$ as $2^{2} \cdot 23$ .

Factor $4$ out of $92$ .

$y = \frac{2 \pm i \cdot \sqrt{4 (23)}}{2 \cdot 3}$

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

$y = \frac{2 \pm i \cdot \sqrt{2^{2} \cdot 23}}{2 \cdot 3}$

Pull terms out from under the radical.

$y = \frac{2 \pm i \cdot (2 \sqrt{23})}{2 \cdot 3}$

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

$y = \frac{2 \pm 2 i \sqrt{23}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$y = \frac{2 \pm 2 i \sqrt{23}}{6}$

Simplify $\frac{2 \pm 2 i \sqrt{23}}{6}$ .

$y = \frac{1 \pm i \sqrt{23}}{3}$

Step 6

The final answer is the combination of both solutions.

$y = \frac{1 + i \sqrt{23}}{3}, \frac{1 - i \sqrt{23}}{3}$

Solve for y 2+(3y-2)y=-6

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