Solve for v 10-16v=-6v^2

$10 - 16 v = - 6 v^{2}$

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

$10 - 16 v + 6 v^{2} = 0$

Factor the left side of the equation.

Factor $2$ out of $10 - 16 v + 6 v^{2}$ .

Factor $2$ out of $10$ .

$2 (5) - 16 v + 6 v^{2} = 0$

Factor $2$ out of $- 16 v$ .

$2 (5) + 2 (- 8 v) + 6 v^{2} = 0$

Factor $2$ out of $6 v^{2}$ .

$2 (5) + 2 (- 8 v) + 2 (3 v^{2}) = 0$

Factor $2$ out of $2 (5) + 2 (- 8 v)$ .

$2 (5 - 8 v) + 2 (3 v^{2}) = 0$

Factor $2$ out of $2 (5 - 8 v) + 2 (3 v^{2})$ .

$2 (5 - 8 v + 3 v^{2}) = 0$

Factor.

Factor by grouping.

Reorder terms.

$2 (3 v^{2} - 8 v + 5) = 0$

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 $a \cdot c = 3 \cdot 5 = 15$ and whose sum is $b = - 8$ .

Factor $- 8$ out of $- 8 v$ .

$2 (3 v^{2} - 8 v + 5) = 0$

Rewrite $- 8$ as $- 3$ plus $- 5$

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

Apply the distributive property.

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

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

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

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

$2 (3 v (v - 1) - 5 (v - 1)) = 0$

Factor the polynomial by factoring out the greatest common factor, $v - 1$ .

$2 ((v - 1) (3 v - 5)) = 0$

Remove unnecessary parentheses.

$2 (v - 1) (3 v - 5) = 0$

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

$v - 1 = 0$

$3 v - 5 = 0$

Set $v - 1$ equal to $0$ and solve for $v$ .

Set $v - 1$ equal to $0$ .

$v - 1 = 0$

Add $1$ to both sides of the equation.

$v = 1$

Set $3 v - 5$ equal to $0$ and solve for $v$ .

Set $3 v - 5$ equal to $0$ .

$3 v - 5 = 0$

Solve $3 v - 5 = 0$ for $v$ .

Add $5$ to 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 $1$ .

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

The final solution is all the values that make $2 (v - 1) (3 v - 5) = 0$ true.

$v = 1, \frac{5}{3}$

The result can be shown in multiple forms.

Exact Form:

$v = 1, \frac{5}{3}$

Decimal Form:

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

Mixed Number Form:

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

Solve for v 10-16v=-6v^2

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