Solve for v -v^2+24v+16=-6v^2

$- v^{2} + 24 v + 16 = - 6 v^{2}$

Step 1

Move all terms containing $v$ to the left side of the equation.

Add $6 v^{2}$ to both sides of the equation.

$- v^{2} + 24 v + 16 + 6 v^{2} = 0$

Add $- v^{2}$ and $6 v^{2}$ .

$5 v^{2} + 24 v + 16 = 0$

Step 2

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 5 \cdot 16 = 80$ and whose sum is $b = 24$ .

Factor $24$ out of $24 v$ .

$5 v^{2} + 24 (v) + 16 = 0$

Rewrite $24$ as $4$ plus $20$

$5 v^{2} + (4 + 20) v + 16 = 0$

Apply the distributive property.

$5 v^{2} + 4 v + 20 v + 16 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(5 v^{2} + 4 v) + 20 v + 16 = 0$

Factor out the greatest common factor (GCF) from each group.

$v (5 v + 4) + 4 (5 v + 4) = 0$

Factor the polynomial by factoring out the greatest common factor, $5 v + 4$ .

$(5 v + 4) (v + 4) = 0$

Step 3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$5 v + 4 = 0$

$v + 4 = 0$

Step 4

Set $5 v + 4$ equal to $0$ and solve for $v$ .

Set $5 v + 4$ equal to $0$ .

$5 v + 4 = 0$

Solve $5 v + 4 = 0$ for $v$ .

Subtract $4$ from both sides of the equation.

$5 v = - 4$

Divide each term in $5 v = - 4$ by $5$ and simplify.

Divide each term in $5 v = - 4$ by $5$ .

$\frac{5 v}{5} = \frac{- 4}{5}$

Simplify the left side.

Cancel the common factor of $5$ .

Cancel the common factor.

$\frac{5 v}{5} = \frac{- 4}{5}$

Divide $v$ by $1$ .

$v = \frac{- 4}{5}$

Simplify the right side.

Move the negative in front of the fraction.

$v = - \frac{4}{5}$

Step 5

Set $v + 4$ equal to $0$ and solve for $v$ .

Set $v + 4$ equal to $0$ .

$v + 4 = 0$

Subtract $4$ from both sides of the equation.

$v = - 4$

Step 6

The final solution is all the values that make $(5 v + 4) (v + 4) = 0$ true.

$v = - \frac{4}{5}, - 4$

Step 7

The result can be shown in multiple forms.

Exact Form:

$v = - \frac{4}{5}, - 4$

Decimal Form:

$v = - 0.8, - 4$

Solve for v -v^2+24v+16=-6v^2

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