Basic Math Examples

$D = c + \frac{v}{a} \cdot (f o r v)$

Step 1

Rewrite the equation as $c + \frac{v}{a} \cdot (f o r v) = D$ .

$c + \frac{v}{a} \cdot (f o r v) = D$

Step 2

Multiply $\frac{v}{a} (f o r v)$ .

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Step 2.1

Combine $f$ and $\frac{v}{a}$ .

$c + \frac{f v}{a} (o r v) = D$

Step 2.2

Combine $o$ and $\frac{f v}{a}$ .

$c + \frac{o (f v)}{a} (r v) = D$

Step 2.3

Combine $r$ and $\frac{o (f v)}{a}$ .

$c + \frac{r (o (f v))}{a} v = D$

Step 2.4

Combine $\frac{r (o (f v))}{a}$ and $v$ .

$c + \frac{r (o (f v)) v}{a} = D$

Step 2.5

Raise $v$ to the power of $1$ .

$c + \frac{r (o (f (v^{1} v)))}{a} = D$

Step 2.6

Raise $v$ to the power of $1$ .

$c + \frac{r (o (f (v^{1} v^{1})))}{a} = D$

Step 2.7

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

$c + \frac{r (o (f v^{1 + 1}))}{a} = D$

Step 2.8

Add $1$ and $1$ .

$c + \frac{r (o (f v^{2}))}{a} = D$

$c + \frac{r o f v^{2}}{a} = D$

Step 3

Subtract $c$ from both sides of the equation.

$\frac{r o f v^{2}}{a} = D - c$

Step 4

Multiply both sides by $a$ .

$\frac{r o f v^{2}}{a} a = (D - c) a$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Cancel the common factor of $a$ .

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Step 5.1.1.1

Cancel the common factor.

$\frac{r o f v^{2}}{a} a = (D - c) a$

Step 5.1.1.2

Rewrite the expression.

$r o f v^{2} = (D - c) a$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $(D - c) a$ .

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Step 5.2.1.1

Apply the distributive property.

$r o f v^{2} = D a - c a$

Step 5.2.1.2

Move $c$ .

$r o f v^{2} = D a - 1 a c$

Step 6

Solve for $v$ .

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Step 6.1

Divide each term in $r o f v^{2} = D a - 1 a c$ by $r o f$ and simplify.

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Step 6.1.1

Divide each term in $r o f v^{2} = D a - 1 a c$ by $r o f$ .

$\frac{r o f v^{2}}{r o f} = \frac{D a}{r o f} + \frac{- 1 a c}{r o f}$

Step 6.1.2

Simplify the left side.

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Step 6.1.2.1

Cancel the common factor of $r$ .

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Step 6.1.2.1.1

Cancel the common factor.

$\frac{r o f v^{2}}{r o f} = \frac{D a}{r o f} + \frac{- 1 a c}{r o f}$

Step 6.1.2.1.2

Rewrite the expression.

$\frac{o f v^{2}}{o f} = \frac{D a}{r o f} + \frac{- 1 a c}{r o f}$

Step 6.1.2.2

Cancel the common factor of $o$ .

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Step 6.1.2.2.1

Cancel the common factor.

$\frac{o f v^{2}}{o f} = \frac{D a}{r o f} + \frac{- 1 a c}{r o f}$

Step 6.1.2.2.2

Rewrite the expression.

$\frac{f v^{2}}{f} = \frac{D a}{r o f} + \frac{- 1 a c}{r o f}$

Step 6.1.2.3

Cancel the common factor of $f$ .

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Step 6.1.2.3.1

Cancel the common factor.

$\frac{f v^{2}}{f} = \frac{D a}{r o f} + \frac{- 1 a c}{r o f}$

Step 6.1.2.3.2

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

$v^{2} = \frac{D a}{r o f} + \frac{- 1 a c}{r o f}$

Step 6.1.3

Simplify the right side.

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Step 6.1.3.1

Move the negative in front of the fraction.

$v^{2} = \frac{D a}{r o f} - \frac{a c}{r o f}$

Step 6.2

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

$v = \pm \sqrt{\frac{D a}{r o f} - \frac{a c}{r o f}}$

Step 6.3

Simplify $\pm \sqrt{\frac{D a}{r o f} - \frac{a c}{r o f}}$ .

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Step 6.3.1

Combine the numerators over the common denominator.

$v = \pm \sqrt{\frac{D a - a c}{r o f}}$

Step 6.3.2

Simplify the numerator.

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Step 6.3.2.1

Factor $a$ out of $D a - a c$ .

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Step 6.3.2.1.1

Factor $a$ out of $D a$ .

$v = \pm \sqrt{\frac{a D - a c}{r o f}}$

Step 6.3.2.1.2

Factor $a$ out of $- a c$ .

$v = \pm \sqrt{\frac{a D + a (- 1 c)}{r o f}}$

Step 6.3.2.1.3

Factor $a$ out of $a D + a (- 1 c)$ .

$v = \pm \sqrt{\frac{a (D - 1 c)}{r o f}}$

Step 6.3.2.2

Rewrite $- 1 c$ as $- c$ .

$v = \pm \sqrt{\frac{a (D - c)}{r o f}}$

Step 6.3.3

Rewrite $\sqrt{\frac{a (D - c)}{r o f}}$ as $\frac{\sqrt{a (D - c)}}{\sqrt{r o f}}$ .

$v = \pm \frac{\sqrt{a (D - c)}}{\sqrt{r o f}}$

Step 6.3.4

Multiply $\frac{\sqrt{a (D - c)}}{\sqrt{r o f}}$ by $\frac{\sqrt{r o f}}{\sqrt{r o f}}$ .

$v = \pm \frac{\sqrt{a (D - c)}}{\sqrt{r o f}} \cdot \frac{\sqrt{r o f}}{\sqrt{r o f}}$

Step 6.3.5

Combine and simplify the denominator.

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Step 6.3.5.1

Multiply $\frac{\sqrt{a (D - c)}}{\sqrt{r o f}}$ by $\frac{\sqrt{r o f}}{\sqrt{r o f}}$ .

$v = \pm \frac{\sqrt{a (D - c)} \sqrt{r o f}}{\sqrt{r o f} \sqrt{r o f}}$

Step 6.3.5.2

Raise $\sqrt{r o f}$ to the power of $1$ .

$v = \pm \frac{\sqrt{a (D - c)} \sqrt{r o f}}{{\sqrt{r o f}}^{1} \sqrt{r o f}}$

Step 6.3.5.3

Raise $\sqrt{r o f}$ to the power of $1$ .

$v = \pm \frac{\sqrt{a (D - c)} \sqrt{r o f}}{{\sqrt{r o f}}^{1} {\sqrt{r o f}}^{1}}$

Step 6.3.5.4

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

$v = \pm \frac{\sqrt{a (D - c)} \sqrt{r o f}}{{\sqrt{r o f}}^{1 + 1}}$

Step 6.3.5.5

Add $1$ and $1$ .

$v = \pm \frac{\sqrt{a (D - c)} \sqrt{r o f}}{{\sqrt{r o f}}^{2}}$

Step 6.3.5.6

Rewrite ${\sqrt{r o f}}^{2}$ as $r o f$ .

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Step 6.3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{r o f}$ as ${(r o f)}^{\frac{1}{2}}$ .

$v = \pm \frac{\sqrt{a (D - c)} \sqrt{r o f}}{{({(r o f)}^{\frac{1}{2}})}^{2}}$

Step 6.3.5.6.2

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

$v = \pm \frac{\sqrt{a (D - c)} \sqrt{r o f}}{{(r o f)}^{\frac{1}{2} \cdot 2}}$

Step 6.3.5.6.3

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

$v = \pm \frac{\sqrt{a (D - c)} \sqrt{r o f}}{{(r o f)}^{\frac{2}{2}}}$

Step 6.3.5.6.4

Cancel the common factor of $2$ .

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Step 6.3.5.6.4.1

Cancel the common factor.

$v = \pm \frac{\sqrt{a (D - c)} \sqrt{r o f}}{{(r o f)}^{\frac{2}{2}}}$

Step 6.3.5.6.4.2

Rewrite the expression.

$v = \pm \frac{\sqrt{a (D - c)} \sqrt{r o f}}{{(r o f)}^{1}}$

Step 6.3.5.6.5

Simplify.

$v = \pm \frac{\sqrt{a (D - c)} \sqrt{r o f}}{r o f}$

Step 6.3.6

Combine using the product rule for radicals.

$v = \pm \frac{\sqrt{a (D - c) r o f}}{r o f}$

Step 6.3.7

Reorder factors in $\pm \frac{\sqrt{a (D - c) r o f}}{r o f}$ .

$v = \pm \frac{\sqrt{a r o f (D - c)}}{r o f}$

Step 6.4

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

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Step 6.4.1

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

$v = \frac{\sqrt{a r o f (D - c)}}{r o f}$

Step 6.4.2

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

$v = - \frac{\sqrt{a r o f (D - c)}}{r o f}$

Step 6.4.3

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

$v = \frac{\sqrt{a r o f (D - c)}}{r o f}$

$v = - \frac{\sqrt{a r o f (D - c)}}{r o f}$

$v = \frac{\sqrt{a r o f (D - c)}}{r o f}$

$v = - \frac{\sqrt{a r o f (D - c)}}{r o f}$

$v = \frac{\sqrt{a r o f (D - c)}}{r o f}$

$v = - \frac{\sqrt{a r o f (D - c)}}{r o f}$