Solve for y y(2y+1)+10=0

$y (2 y + 1) + 10 = 0$

Step 1

Simplify each term.

Apply the distributive property.

$y (2 y) + y \cdot 1 + 10 = 0$

Rewrite using the commutative property of multiplication.

$2 y \cdot y + y \cdot 1 + 10 = 0$

Multiply $y$ by $1$ .

$2 y \cdot y + y + 10 = 0$

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

Move $y$ .

$2 (y \cdot y) + y + 10 = 0$

Multiply $y$ by $y$ .

$2 y^{2} + y + 10 = 0$

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 $a = 2$ , $b = 1$ , and $c = 10$ into the quadratic formula and solve for $y$ .

$\frac{- 1 \pm \sqrt{1^{2} - 4 \cdot (2 \cdot 10)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

One to any power is one.

$y = \frac{- 1 \pm \sqrt{1 - 4 \cdot 2 \cdot 10}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 10$ .

Multiply $- 4$ by $2$ .

$y = \frac{- 1 \pm \sqrt{1 - 8 \cdot 10}}{2 \cdot 2}$

Multiply $- 8$ by $10$ .

$y = \frac{- 1 \pm \sqrt{1 - 80}}{2 \cdot 2}$

Subtract $80$ from $1$ .

$y = \frac{- 1 \pm \sqrt{- 79}}{2 \cdot 2}$

Rewrite $- 79$ as $- 1 (79)$ .

$y = \frac{- 1 \pm \sqrt{- 1 \cdot 79}}{2 \cdot 2}$

Rewrite $\sqrt{- 1 (79)}$ as $\sqrt{- 1} \cdot \sqrt{79}$ .

$y = \frac{- 1 \pm \sqrt{- 1} \cdot \sqrt{79}}{2 \cdot 2}$

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

$y = \frac{- 1 \pm i \sqrt{79}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 1 \pm i \sqrt{79}}{4}$

Step 5

The final answer is the combination of both solutions.

$y = - \frac{1 - i \sqrt{79}}{4}, - \frac{1 + i \sqrt{79}}{4}$