Algebra Examples

$v e = \frac{1}{2} m v^{2}$

Step 1

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

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Step 1.1

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

$v e = \frac{m}{2} v^{2}$

Step 1.2

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

$v e = \frac{m v^{2}}{2}$

Step 2

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

$v e - \frac{m v^{2}}{2} = 0$

Step 3

Factor $v$ out of $v e - \frac{m v^{2}}{2}$ .

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Step 3.1

Factor $v$ out of $v e$ .

$v (e) - \frac{m v^{2}}{2} = 0$

Step 3.2

Factor $v$ out of $- \frac{m v^{2}}{2}$ .

$v (e) + v (- \frac{m v}{2}) = 0$

Step 3.3

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

$v (e - \frac{m v}{2}) = 0$

Step 4

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

$v = 0$

$e - \frac{m v}{2} = 0$

Step 5

Set $v$ equal to $0$ .

$v = 0$

Step 6

Set $e - \frac{m v}{2}$ equal to $0$ and solve for $v$ .

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Step 6.1

Set $e - \frac{m v}{2}$ equal to $0$ .

$e - \frac{m v}{2} = 0$

Step 6.2

Solve $e - \frac{m v}{2} = 0$ for $v$ .

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Step 6.2.1

Subtract $e$ from both sides of the equation.

$- \frac{m v}{2} = - e$

Step 6.2.2

Multiply both sides of the equation by $- 2$ .

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

Step 6.2.3

Simplify both sides of the equation.

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Step 6.2.3.1

Simplify the left side.

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Step 6.2.3.1.1

Simplify $- 2 (- \frac{m v}{2})$ .

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Step 6.2.3.1.1.1

Cancel the common factor of $2$ .

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Step 6.2.3.1.1.1.1

Move the leading negative in $- \frac{m v}{2}$ into the numerator.

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

Step 6.2.3.1.1.1.2

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{- m v}{2} = - 2 (- e)$

Step 6.2.3.1.1.1.3

Cancel the common factor.

$2 \cdot - 1 \frac{- m v}{2} = - 2 (- e)$

Step 6.2.3.1.1.1.4

Rewrite the expression.

$- (- m v) = - 2 (- e)$

Step 6.2.3.1.1.2

Multiply.

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Step 6.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 (m v) = - 2 (- e)$

Step 6.2.3.1.1.2.2

Multiply $m$ by $1$ .

$m v = - 2 (- e)$

Step 6.2.3.2

Simplify the right side.

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Step 6.2.3.2.1

Multiply $- 1$ by $- 2$ .

$m v = 2 e$

Step 6.2.4

Divide each term in $m v = 2 e$ by $m$ and simplify.

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Step 6.2.4.1

Divide each term in $m v = 2 e$ by $m$ .

$\frac{m v}{m} = \frac{2 e}{m}$

Step 6.2.4.2

Simplify the left side.

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Step 6.2.4.2.1

Cancel the common factor of $m$ .

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Step 6.2.4.2.1.1

Cancel the common factor.

$\frac{m v}{m} = \frac{2 e}{m}$

Step 6.2.4.2.1.2

Divide $v$ by $1$ .

$v = \frac{2 e}{m}$

Step 7

The final solution is all the values that make $v (e - \frac{m v}{2}) = 0$ true.

$v = 0$

$v = \frac{2 e}{m}$