Solve for v v = square root of -18+9v

$v = \sqrt{- 18 + 9 v}$

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{- 18 + 9 v} = v$

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{- 18 + 9 v}}^{2} = v^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{- 18 + 9 v}$ as ${(- 18 + 9 v)}^{\frac{1}{2}}$ .

${({(- 18 + 9 v)}^{\frac{1}{2}})}^{2} = v^{2}$

Simplify the left side.

Simplify ${({(- 18 + 9 v)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(- 18 + 9 v)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(- 18 + 9 v)}^{\frac{1}{2} \cdot 2} = v^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(- 18 + 9 v)}^{\frac{1}{2} \cdot 2} = v^{2}$

Rewrite the expression.

${(- 18 + 9 v)}^{1} = v^{2}$

Simplify.

$- 18 + 9 v = v^{2}$

Solve for $v$ .

Subtract $v^{2}$ from both sides of the equation.

$- 18 + 9 v - v^{2} = 0$

Factor the left side of the equation.

Factor $- 1$ out of $- 18 + 9 v - v^{2}$ .

Reorder the expression.

Move $- 18$ .

$9 v - v^{2} - 18 = 0$

Reorder $9 v$ and $- v^{2}$ .

$- v^{2} + 9 v - 18 = 0$

Factor $- 1$ out of $- v^{2}$ .

$- (v^{2}) + 9 v - 18 = 0$

Factor $- 1$ out of $9 v$ .

$- (v^{2}) - (- 9 v) - 18 = 0$

Rewrite $- 18$ as $- 1 (18)$ .

$- (v^{2}) - (- 9 v) - 1 \cdot 18 = 0$

Factor $- 1$ out of $- (v^{2}) - (- 9 v)$ .

$- (v^{2} - 9 v) - 1 \cdot 18 = 0$

Factor $- 1$ out of $- (v^{2} - 9 v) - 1 (18)$ .

$- (v^{2} - 9 v + 18) = 0$

Factor.

Factor $v^{2} - 9 v + 18$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $18$ and whose sum is $- 9$ .

$- 6, - 3$

Write the factored form using these integers.

$- ((v - 6) (v - 3)) = 0$

Remove unnecessary parentheses.

$- (v - 6) (v - 3) = 0$

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

$v - 6 = 0$

$v - 3 = 0$

Set $v - 6$ equal to $0$ and solve for $v$ .

Set $v - 6$ equal to $0$ .

$v - 6 = 0$

Add $6$ to both sides of the equation.

$v = 6$

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

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

$v - 3 = 0$

Add $3$ to both sides of the equation.

$v = 3$

The final solution is all the values that make $- (v - 6) (v - 3) = 0$ true.

$v = 6, 3$

Solve for v v = square root of -18+9v

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