Algebra Examples

$E = \frac{1}{2} m v^{2} + mg h$

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Multiply both sides of the equation by $2$ .

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

Step 5

Simplify both sides of the equation.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $2$ .

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Step 5.1.1.1

Cancel the common factor.

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

Step 5.1.1.2

Rewrite the expression.

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

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $2 (E - mg h)$ .

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Step 5.2.1.1

Apply the distributive property.

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

Step 5.2.1.2

Multiply $- 1$ by $2$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $m$ .

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Step 6.2.1.1

Cancel the common factor.

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

Step 6.2.1.2

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

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

Step 6.3

Simplify the right side.

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Step 6.3.1

Move the negative in front of the fraction.

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

Step 7

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

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

Step 8

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

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Step 8.1

Combine the numerators over the common denominator.

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

Step 8.2

Factor $2$ out of $2 E - 2 mg h$ .

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Step 8.2.1

Factor $2$ out of $2 E$ .

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

Step 8.2.2

Factor $2$ out of $- 2 mg h$ .

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

Step 8.2.3

Factor $2$ out of $2 (E) + 2 (- mg h)$ .

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Combine and simplify the denominator.

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Step 8.5.1

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

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

Step 8.5.2

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

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

Step 8.5.3

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

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

Step 8.5.4

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

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

Step 8.5.5

Add $1$ and $1$ .

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

Step 8.5.6

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

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Step 8.5.6.1

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

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

Step 8.5.6.2

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

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

Step 8.5.6.3

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

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

Step 8.5.6.4

Cancel the common factor of $2$ .

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Step 8.5.6.4.1

Cancel the common factor.

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

Step 8.5.6.4.2

Rewrite the expression.

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

Step 8.5.6.5

Simplify.

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

Step 8.6

Combine using the product rule for radicals.

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

Step 8.7

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

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

Step 9

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

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 10

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

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