Solve for r x=qV+1/2mr^2

$x = q V + \frac{1}{2} m r^{2}$

Rewrite the equation as $q V + \frac{1}{2} \cdot (m r^{2}) = x$ .

$q V + \frac{1}{2} \cdot (m r^{2}) = x$

Multiply $\frac{1}{2} (m r^{2})$ .

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

$q V + \frac{m}{2} r^{2} = x$

Combine $\frac{m}{2}$ and $r^{2}$ .

$q V + \frac{m r^{2}}{2} = x$

Subtract $q V$ from both sides of the equation.

$\frac{m r^{2}}{2} = x - q V$

Multiply both sides of the equation by $2$ .

$2 \frac{m r^{2}}{2} = 2 (x - q V)$

Simplify both sides of the equation.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$2 \frac{m r^{2}}{2} = 2 (x - q V)$

Rewrite the expression.

$m r^{2} = 2 (x - q V)$

Simplify the right side.

Simplify $2 (x - q V)$ .

Apply the distributive property.

$m r^{2} = 2 x + 2 (- q V)$

Multiply $- 1$ by $2$ .

$m r^{2} = 2 x - 2 q V$

Divide each term in $m r^{2} = 2 x - 2 q V$ by $m$ and simplify.

Divide each term in $m r^{2} = 2 x - 2 q V$ by $m$ .

$\frac{m r^{2}}{m} = \frac{2 x}{m} + \frac{- 2 q V}{m}$

Simplify the left side.

Cancel the common factor of $m$ .

Cancel the common factor.

$\frac{m r^{2}}{m} = \frac{2 x}{m} + \frac{- 2 q V}{m}$

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

$r^{2} = \frac{2 x}{m} + \frac{- 2 q V}{m}$

Simplify the right side.

Move the negative in front of the fraction.

$r^{2} = \frac{2 x}{m} - \frac{2 q V}{m}$

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

$r = \pm \sqrt{\frac{2 x}{m} - \frac{2 q V}{m}}$

Simplify $\pm \sqrt{\frac{2 x}{m} - \frac{2 q V}{m}}$ .

Combine the numerators over the common denominator.

$r = \pm \sqrt{\frac{2 x - 2 q V}{m}}$

Factor $2$ out of $2 x - 2 q V$ .

Factor $2$ out of $2 x$ .

$r = \pm \sqrt{\frac{2 (x) - 2 q V}{m}}$

Factor $2$ out of $- 2 q V$ .

$r = \pm \sqrt{\frac{2 (x) + 2 (- q V)}{m}}$

Factor $2$ out of $2 (x) + 2 (- q V)$ .

$r = \pm \sqrt{\frac{2 (x - q V)}{m}}$

Rewrite $\sqrt{\frac{2 (x - q V)}{m}}$ as $\frac{\sqrt{2 (x - q V)}}{\sqrt{m}}$ .

$r = \pm \frac{\sqrt{2 (x - q V)}}{\sqrt{m}}$

Multiply $\frac{\sqrt{2 (x - q V)}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$r = \pm \frac{\sqrt{2 (x - q V)}}{\sqrt{m}} \cdot \frac{\sqrt{m}}{\sqrt{m}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{2 (x - q V)}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{\sqrt{m} \sqrt{m}}$

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

$r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{{\sqrt{m}}^{1} \sqrt{m}}$

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

$r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{{\sqrt{m}}^{1} {\sqrt{m}}^{1}}$

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

$r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{{\sqrt{m}}^{1 + 1}}$

Add $1$ and $1$ .

$r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{{\sqrt{m}}^{2}}$

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

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

$r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{{(m^{\frac{1}{2}})}^{2}}$

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

$r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{m^{\frac{1}{2} \cdot 2}}$

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

$r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{m^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{m^{\frac{2}{2}}}$

Rewrite the expression.

$r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{m^{1}}$

Simplify.

$r = \pm \frac{\sqrt{2 (x - q V)} \sqrt{m}}{m}$

Combine using the product rule for radicals.

$r = \pm \frac{\sqrt{2 (x - q V) m}}{m}$

Reorder factors in $\pm \frac{\sqrt{2 (x - q V) m}}{m}$ .

$r = \pm \frac{\sqrt{2 m (x - q V)}}{m}$

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.

$r = \frac{\sqrt{2 m (x - q V)}}{m}$

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

$r = - \frac{\sqrt{2 m (x - q V)}}{m}$

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

$r = \frac{\sqrt{2 m (x - q V)}}{m}$

$r = - \frac{\sqrt{2 m (x - q V)}}{m}$

$r = \frac{\sqrt{2 m (x - q V)}}{m}$

$r = - \frac{\sqrt{2 m (x - q V)}}{m}$

Solve for r x=qV+1/2mr^2

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