Algebra Examples

$m e = \frac{1}{2} \cdot (m v^{2}) + mg h$

Step 1

Rewrite the equation as $\frac{1}{2} \cdot (m v^{2}) + mg h = m e$ .

$\frac{1}{2} \cdot (m v^{2}) + mg h = m e$

Step 2

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

Tap for more steps...

Step 2.1

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

$\frac{m}{2} v^{2} + mg h = m e$

Step 2.2

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

$\frac{m v^{2}}{2} + mg h = m e$

Step 3

Subtract $mg h$ from both sides of the equation.

$\frac{m v^{2}}{2} = m e - mg h$

Step 4

Multiply both sides of the equation by $2$ .

$2 \frac{m v^{2}}{2} = 2 (m e - mg h)$

Step 5

Simplify both sides of the equation.

Tap for more steps...

Step 5.1

Simplify the left side.

Tap for more steps...

Step 5.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 5.1.1.1

Cancel the common factor.

$2 \frac{m v^{2}}{2} = 2 (m e - mg h)$

Step 5.1.1.2

Rewrite the expression.

$m v^{2} = 2 (m e - mg h)$

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Simplify $2 (m e - mg h)$ .

Tap for more steps...

Step 5.2.1.1

Apply the distributive property.

$m v^{2} = 2 (m e) + 2 (- mg h)$

Step 5.2.1.2

Multiply $- 1$ by $2$ .

$m v^{2} = 2 m e - 2 mg h$

Step 6

Divide each term in $m v^{2} = 2 m e - 2 mg h$ by $m$ and simplify.

Tap for more steps...

Step 6.1

Divide each term in $m v^{2} = 2 m e - 2 mg h$ by $m$ .

$\frac{m v^{2}}{m} = \frac{2 m e}{m} + \frac{- 2 mg h}{m}$

Step 6.2

Simplify the left side.

Tap for more steps...

Step 6.2.1

Cancel the common factor of $m$ .

Tap for more steps...

Step 6.2.1.1

Cancel the common factor.

$\frac{m v^{2}}{m} = \frac{2 m e}{m} + \frac{- 2 mg h}{m}$

Step 6.2.1.2

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

$v^{2} = \frac{2 m e}{m} + \frac{- 2 mg h}{m}$

Step 6.3

Simplify the right side.

Tap for more steps...

Step 6.3.1

Simplify each term.

Tap for more steps...

Step 6.3.1.1

Cancel the common factor of $m$ .

Tap for more steps...

Step 6.3.1.1.1

Cancel the common factor.

$v^{2} = \frac{2 m e}{m} + \frac{- 2 mg h}{m}$

Step 6.3.1.1.2

Divide $2 e$ by $1$ .

$v^{2} = 2 e + \frac{- 2 mg h}{m}$

Step 6.3.1.2

Move the negative in front of the fraction.

$v^{2} = 2 e - \frac{2 mg h}{m}$

Step 7

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

$v = \pm \sqrt{2 e - \frac{2 mg h}{m}}$

Step 8

Simplify $\pm \sqrt{2 e - \frac{2 mg h}{m}}$ .

Tap for more steps...

Step 8.1

Factor $2$ out of $2 e - \frac{2 mg h}{m}$ .

Tap for more steps...

Step 8.1.1

Factor $2$ out of $2 e$ .

$v = \pm \sqrt{2 (e) - \frac{2 mg h}{m}}$

Step 8.1.2

Factor $2$ out of $- \frac{2 mg h}{m}$ .

$v = \pm \sqrt{2 (e) + 2 (- \frac{mg h}{m})}$

Step 8.1.3

Factor $2$ out of $2 (e) + 2 (- \frac{mg h}{m})$ .

$v = \pm \sqrt{2 (e - \frac{mg h}{m})}$

Step 8.2

To write $e$ as a fraction with a common denominator, multiply by $\frac{m}{m}$ .

$v = \pm \sqrt{2 (\frac{e m}{m} - \frac{mg h}{m})}$

Step 8.3

Combine the numerators over the common denominator.

$v = \pm \sqrt{2 \frac{e m - mg h}{m}}$

Step 8.4

Combine $2$ and $\frac{e m - mg h}{m}$ .

$v = \pm \sqrt{\frac{2 (e m - mg h)}{m}}$

Step 8.5

Rewrite $\sqrt{\frac{2 (e m - mg h)}{m}}$ as $\frac{\sqrt{2 (e m - mg h)}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 (e m - mg h)}}{\sqrt{m}}$

Step 8.6

Multiply $\frac{\sqrt{2 (e m - mg h)}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 (e m - mg h)}}{\sqrt{m}} \cdot \frac{\sqrt{m}}{\sqrt{m}}$

Step 8.7

Combine and simplify the denominator.

Tap for more steps...

Step 8.7.1

Multiply $\frac{\sqrt{2 (e m - mg h)}}{\sqrt{m}}$ by $\frac{\sqrt{m}}{\sqrt{m}}$ .

$v = \pm \frac{\sqrt{2 (e m - mg h)} \sqrt{m}}{\sqrt{m} \sqrt{m}}$

Step 8.7.2

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

$v = \pm \frac{\sqrt{2 (e m - mg h)} \sqrt{m}}{{\sqrt{m}}^{1} \sqrt{m}}$

Step 8.7.3

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

$v = \pm \frac{\sqrt{2 (e m - mg h)} \sqrt{m}}{{\sqrt{m}}^{1} {\sqrt{m}}^{1}}$

Step 8.7.4

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

$v = \pm \frac{\sqrt{2 (e m - mg h)} \sqrt{m}}{{\sqrt{m}}^{1 + 1}}$

Step 8.7.5

Add $1$ and $1$ .

$v = \pm \frac{\sqrt{2 (e m - mg h)} \sqrt{m}}{{\sqrt{m}}^{2}}$

Step 8.7.6

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

Tap for more steps...

Step 8.7.6.1

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

$v = \pm \frac{\sqrt{2 (e m - mg h)} \sqrt{m}}{{(m^{\frac{1}{2}})}^{2}}$

Step 8.7.6.2

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

$v = \pm \frac{\sqrt{2 (e m - mg h)} \sqrt{m}}{m^{\frac{1}{2} \cdot 2}}$

Step 8.7.6.3

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

$v = \pm \frac{\sqrt{2 (e m - mg h)} \sqrt{m}}{m^{\frac{2}{2}}}$

Step 8.7.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.7.6.4.1

Cancel the common factor.

$v = \pm \frac{\sqrt{2 (e m - mg h)} \sqrt{m}}{m^{\frac{2}{2}}}$

Step 8.7.6.4.2

Rewrite the expression.

$v = \pm \frac{\sqrt{2 (e m - mg h)} \sqrt{m}}{m^{1}}$

Step 8.7.6.5

Simplify.

$v = \pm \frac{\sqrt{2 (e m - mg h)} \sqrt{m}}{m}$

Step 8.8

Combine using the product rule for radicals.

$v = \pm \frac{\sqrt{2 (e m - mg h) m}}{m}$

Step 8.9

Reorder factors in $\pm \frac{\sqrt{2 (e m - mg h) m}}{m}$ .

$v = \pm \frac{\sqrt{2 m (e m - mg h)}}{m}$

Step 9

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

Tap for more steps...

Step 9.1

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

$v = \frac{\sqrt{2 m (e m - mg h)}}{m}$

Step 9.2

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

$v = - \frac{\sqrt{2 m (e m - mg h)}}{m}$

Step 9.3

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

$v = \frac{\sqrt{2 m (e m - mg h)}}{m}, - \frac{\sqrt{2 m (e m - mg h)}}{m}$

Step 10

Reorder factors in $v = \frac{\sqrt{2 m (e m - mg h)}}{m}, - \frac{\sqrt{2 m (e m - mg h)}}{m}$ .

$v = \frac{\sqrt{2 m (e m - h mg)}}{m}, - \frac{\sqrt{2 m (e m - h mg)}}{m}$