Solve Using the Quadratic Formula 3y^2+6y+5=0

$3 y^{2} + 6 y + 5 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 3$ , $b = 6$ , and $c = 5$ into the quadratic formula and solve for $y$ .

$\frac{- 6 \pm \sqrt{6^{2} - 4 \cdot (3 \cdot 5)}}{2 \cdot 3}$

Step 3

Simplify.

Simplify the numerator.

Raise $6$ to the power of $2$ .

$y = \frac{- 6 \pm \sqrt{36 - 4 \cdot 3 \cdot 5}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 5$ .

Multiply $- 4$ by $3$ .

$y = \frac{- 6 \pm \sqrt{36 - 12 \cdot 5}}{2 \cdot 3}$

Multiply $- 12$ by $5$ .

$y = \frac{- 6 \pm \sqrt{36 - 60}}{2 \cdot 3}$

Subtract $60$ from $36$ .

$y = \frac{- 6 \pm \sqrt{- 24}}{2 \cdot 3}$

Rewrite $- 24$ as $- 1 (24)$ .

$y = \frac{- 6 \pm \sqrt{- 1 \cdot 24}}{2 \cdot 3}$

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$y = \frac{- 6 \pm \sqrt{- 1} \cdot \sqrt{24}}{2 \cdot 3}$

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

$y = \frac{- 6 \pm i \cdot \sqrt{24}}{2 \cdot 3}$

Rewrite $24$ as $2^{2} \cdot 6$ .

Factor $4$ out of $24$ .

$y = \frac{- 6 \pm i \cdot \sqrt{4 (6)}}{2 \cdot 3}$

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

$y = \frac{- 6 \pm i \cdot \sqrt{2^{2} \cdot 6}}{2 \cdot 3}$

Pull terms out from under the radical.

$y = \frac{- 6 \pm i \cdot (2 \sqrt{6})}{2 \cdot 3}$

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

$y = \frac{- 6 \pm 2 i \sqrt{6}}{2 \cdot 3}$

Multiply $2$ by $3$ .

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

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

$y = \frac{- 3 \pm i \sqrt{6}}{3}$