Solve for v 9v^2=-30v-25

$9 v^{2} = - 30 v - 25$

Add $30 v$ to both sides of the equation.

$9 v^{2} + 30 v = - 25$

Add $25$ to both sides of the equation.

$9 v^{2} + 30 v + 25 = 0$

Factor using the perfect square rule.

Rewrite $9 v^{2}$ as ${(3 v)}^{2}$ .

${(3 v)}^{2} + 30 v + 25 = 0$

Rewrite $25$ as $5^{2}$ .

${(3 v)}^{2} + 30 v + 5^{2} = 0$

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$30 v = 2 \cdot (3 v) \cdot 5$

Rewrite the polynomial.

${(3 v)}^{2} + 2 \cdot (3 v) \cdot 5 + 5^{2} = 0$

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = 3 v$ and $b = 5$ .

${(3 v + 5)}^{2} = 0$

Set the $3 v + 5$ equal to $0$ .

$3 v + 5 = 0$

Solve for $v$ .

Subtract $5$ from both sides of the equation.

$3 v = - 5$

Divide each term in $3 v = - 5$ by $3$ and simplify.

Divide each term in $3 v = - 5$ by $3$ .

$\frac{3 v}{3} = \frac{- 5}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 v}{3} = \frac{- 5}{3}$

Divide $v$ by $1$ .

$v = \frac{- 5}{3}$

Simplify the right side.

Move the negative in front of the fraction.

$v = - \frac{5}{3}$

The result can be shown in multiple forms.

Exact Form:

$v = - \frac{5}{3}$

Decimal Form:

$v = - 1.\overline{6}$

Mixed Number Form:

$v = - 1 \frac{2}{3}$