Solve for w vw^2+y=x

$v w^{2} + y = x$

Subtract $y$ from both sides of the equation.

$v w^{2} = x - y$

Divide each term in $v w^{2} = x - y$ by $v$ and simplify.

Divide each term in $v w^{2} = x - y$ by $v$ .

$\frac{v w^{2}}{v} = \frac{x}{v} + \frac{- y}{v}$

Simplify the left side.

Cancel the common factor of $v$ .

Cancel the common factor.

$\frac{v w^{2}}{v} = \frac{x}{v} + \frac{- y}{v}$

Divide $w^{2}$ by $1$ .

$w^{2} = \frac{x}{v} + \frac{- y}{v}$

Simplify the right side.

Move the negative in front of the fraction.

$w^{2} = \frac{x}{v} - \frac{y}{v}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$w = \pm \sqrt{\frac{x}{v} - \frac{y}{v}}$

Simplify $\pm \sqrt{\frac{x}{v} - \frac{y}{v}}$ .

Combine the numerators over the common denominator.

$w = \pm \sqrt{\frac{x - y}{v}}$

Rewrite $\sqrt{\frac{x - y}{v}}$ as $\frac{\sqrt{x - y}}{\sqrt{v}}$ .

$w = \pm \frac{\sqrt{x - y}}{\sqrt{v}}$

Multiply $\frac{\sqrt{x - y}}{\sqrt{v}}$ by $\frac{\sqrt{v}}{\sqrt{v}}$ .

$w = \pm \frac{\sqrt{x - y}}{\sqrt{v}} \cdot \frac{\sqrt{v}}{\sqrt{v}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{x - y}}{\sqrt{v}}$ by $\frac{\sqrt{v}}{\sqrt{v}}$ .

$w = \pm \frac{\sqrt{x - y} \sqrt{v}}{\sqrt{v} \sqrt{v}}$

Raise $\sqrt{v}$ to the power of $1$ .

$w = \pm \frac{\sqrt{x - y} \sqrt{v}}{{\sqrt{v}}^{1} \sqrt{v}}$

Raise $\sqrt{v}$ to the power of $1$ .

$w = \pm \frac{\sqrt{x - y} \sqrt{v}}{{\sqrt{v}}^{1} {\sqrt{v}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$w = \pm \frac{\sqrt{x - y} \sqrt{v}}{{\sqrt{v}}^{1 + 1}}$

Add $1$ and $1$ .

$w = \pm \frac{\sqrt{x - y} \sqrt{v}}{{\sqrt{v}}^{2}}$

Rewrite ${\sqrt{v}}^{2}$ as $v$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{v}$ as $v^{\frac{1}{2}}$ .

$w = \pm \frac{\sqrt{x - y} \sqrt{v}}{{(v^{\frac{1}{2}})}^{2}}$

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

$w = \pm \frac{\sqrt{x - y} \sqrt{v}}{v^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$w = \pm \frac{\sqrt{x - y} \sqrt{v}}{v^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$w = \pm \frac{\sqrt{x - y} \sqrt{v}}{v^{\frac{2}{2}}}$

Rewrite the expression.

$w = \pm \frac{\sqrt{x - y} \sqrt{v}}{v^{1}}$

Simplify.

$w = \pm \frac{\sqrt{x - y} \sqrt{v}}{v}$

Combine using the product rule for radicals.

$w = \pm \frac{\sqrt{(x - y) v}}{v}$

Reorder factors in $\pm \frac{\sqrt{(x - y) v}}{v}$ .

$w = \pm \frac{\sqrt{v (x - y)}}{v}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$w = \frac{\sqrt{v (x - y)}}{v}$

Next, use the negative value of the $\pm$ to find the second solution.

$w = - \frac{\sqrt{v (x - y)}}{v}$

The complete solution is the result of both the positive and negative portions of the solution.

$w = \frac{\sqrt{v (x - y)}}{v}$

$w = - \frac{\sqrt{v (x - y)}}{v}$

$w = \frac{\sqrt{v (x - y)}}{v}$

$w = - \frac{\sqrt{v (x - y)}}{v}$

Solve for w vw^2+y=x

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