Solve by Factoring 2y^2+10y=-9

$2 y^{2} + 10 y = - 9$

Step 1

Add $9$ to both sides of the equation.

$2 y^{2} + 10 y + 9 = 0$

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 $a = 2$ , $b = 10$ , and $c = 9$ into the quadratic formula and solve for $y$ .

$\frac{- 10 \pm \sqrt{10^{2} - 4 \cdot (2 \cdot 9)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

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

$y = \frac{- 10 \pm \sqrt{100 - 4 \cdot 2 \cdot 9}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 9$ .

Multiply $- 4$ by $2$ .

$y = \frac{- 10 \pm \sqrt{100 - 8 \cdot 9}}{2 \cdot 2}$

Multiply $- 8$ by $9$ .

$y = \frac{- 10 \pm \sqrt{100 - 72}}{2 \cdot 2}$

Subtract $72$ from $100$ .

$y = \frac{- 10 \pm \sqrt{28}}{2 \cdot 2}$

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$y = \frac{- 10 \pm \sqrt{4 (7)}}{2 \cdot 2}$

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

$y = \frac{- 10 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 10 \pm 2 \sqrt{7}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 10 \pm 2 \sqrt{7}}{4}$

Simplify $\frac{- 10 \pm 2 \sqrt{7}}{4}$ .

$y = \frac{- 5 \pm \sqrt{7}}{2}$

Step 5

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

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

$y = \frac{- 10 \pm \sqrt{100 - 4 \cdot 2 \cdot 9}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 9$ .

Multiply $- 4$ by $2$ .

$y = \frac{- 10 \pm \sqrt{100 - 8 \cdot 9}}{2 \cdot 2}$

Multiply $- 8$ by $9$ .

$y = \frac{- 10 \pm \sqrt{100 - 72}}{2 \cdot 2}$

Subtract $72$ from $100$ .

$y = \frac{- 10 \pm \sqrt{28}}{2 \cdot 2}$

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$y = \frac{- 10 \pm \sqrt{4 (7)}}{2 \cdot 2}$

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

$y = \frac{- 10 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 10 \pm 2 \sqrt{7}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 10 \pm 2 \sqrt{7}}{4}$

Simplify $\frac{- 10 \pm 2 \sqrt{7}}{4}$ .

$y = \frac{- 5 \pm \sqrt{7}}{2}$

Change the $\pm$ to $+$ .

$y = \frac{- 5 + \sqrt{7}}{2}$

Rewrite $- 5$ as $- 1 (5)$ .

$y = \frac{- 1 \cdot 5 + \sqrt{7}}{2}$

Factor $- 1$ out of $\sqrt{7}$ .

$y = \frac{- 1 \cdot 5 - 1 (- \sqrt{7})}{2}$

Factor $- 1$ out of $- 1 (5) - 1 (- \sqrt{7})$ .

$y = \frac{- 1 (5 - \sqrt{7})}{2}$

Move the negative in front of the fraction.

$y = - \frac{5 - \sqrt{7}}{2}$

Step 6

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

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

$y = \frac{- 10 \pm \sqrt{100 - 4 \cdot 2 \cdot 9}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 9$ .

Multiply $- 4$ by $2$ .

$y = \frac{- 10 \pm \sqrt{100 - 8 \cdot 9}}{2 \cdot 2}$

Multiply $- 8$ by $9$ .

$y = \frac{- 10 \pm \sqrt{100 - 72}}{2 \cdot 2}$

Subtract $72$ from $100$ .

$y = \frac{- 10 \pm \sqrt{28}}{2 \cdot 2}$

Rewrite $28$ as $2^{2} \cdot 7$ .

Factor $4$ out of $28$ .

$y = \frac{- 10 \pm \sqrt{4 (7)}}{2 \cdot 2}$

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

$y = \frac{- 10 \pm \sqrt{2^{2} \cdot 7}}{2 \cdot 2}$

Pull terms out from under the radical.

$y = \frac{- 10 \pm 2 \sqrt{7}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$y = \frac{- 10 \pm 2 \sqrt{7}}{4}$

Simplify $\frac{- 10 \pm 2 \sqrt{7}}{4}$ .

$y = \frac{- 5 \pm \sqrt{7}}{2}$

Change the $\pm$ to $-$ .

$y = \frac{- 5 - \sqrt{7}}{2}$

Rewrite $- 5$ as $- 1 (5)$ .

$y = \frac{- 1 \cdot 5 - \sqrt{7}}{2}$

Factor $- 1$ out of $- \sqrt{7}$ .

$y = \frac{- 1 \cdot 5 - (\sqrt{7})}{2}$

Factor $- 1$ out of $- 1 (5) - (\sqrt{7})$ .

$y = \frac{- 1 (5 + \sqrt{7})}{2}$

Move the negative in front of the fraction.

$y = - \frac{5 + \sqrt{7}}{2}$

Step 7

The final answer is the combination of both solutions.

$y = - \frac{5 - \sqrt{7}}{2}, - \frac{5 + \sqrt{7}}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{5 - \sqrt{7}}{2}, - \frac{5 + \sqrt{7}}{2}$

Decimal Form:

$y = - 1.17712434 \dots, - 3.82287565 \dots$

Solve by Factoring 2y^2+10y=-9

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