Basic Math Examples

$0.9 = v (50 - y^{2})$

Step 1

Rewrite the equation as $v (50 - y^{2}) = 0.9$ .

$v (50 - y^{2}) = 0.9$

Step 2

Divide each term in $v (50 - y^{2}) = 0.9$ by $v$ and simplify.

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Step 2.1

Divide each term in $v (50 - y^{2}) = 0.9$ by $v$ .

$\frac{v (50 - y^{2})}{v} = \frac{0.9}{v}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of $v$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{v (50 - y^{2})}{v} = \frac{0.9}{v}$

Step 2.2.1.2

Divide $50 - y^{2}$ by $1$ .

$50 - y^{2} = \frac{0.9}{v}$

Step 3

Subtract $50$ from both sides of the equation.

$- y^{2} = \frac{0.9}{v} - 50$

Step 4

Divide each term in $- y^{2} = \frac{0.9}{v} - 50$ by $- 1$ and simplify.

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Step 4.1

Divide each term in $- y^{2} = \frac{0.9}{v} - 50$ by $- 1$ .

$\frac{- y^{2}}{- 1} = \frac{\frac{0.9}{v}}{- 1} + \frac{- 50}{- 1}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Dividing two negative values results in a positive value.

$\frac{y^{2}}{1} = \frac{\frac{0.9}{v}}{- 1} + \frac{- 50}{- 1}$

Step 4.2.2

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

$y^{2} = \frac{\frac{0.9}{v}}{- 1} + \frac{- 50}{- 1}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify each term.

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Step 4.3.1.1

Move the negative one from the denominator of $\frac{\frac{0.9}{v}}{- 1}$ .

$y^{2} = - 1 \cdot \frac{0.9}{v} + \frac{- 50}{- 1}$

Step 4.3.1.2

Rewrite $- 1 \cdot \frac{0.9}{v}$ as $- \frac{0.9}{v}$ .

$y^{2} = - \frac{0.9}{v} + \frac{- 50}{- 1}$

Step 4.3.1.3

Divide $- 50$ by $- 1$ .

$y^{2} = - \frac{0.9}{v} + 50$

Step 5

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

$y = \pm \sqrt{- \frac{0.9}{v} + 50}$

Step 6

Simplify $\pm \sqrt{- \frac{0.9}{v} + 50}$ .

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Step 6.1

To write $50$ as a fraction with a common denominator, multiply by $\frac{v}{v}$ .

$y = \pm \sqrt{- \frac{0.9}{v} + \frac{50 v}{v}}$

Step 6.2

Combine the numerators over the common denominator.

$y = \pm \sqrt{\frac{- 0.9 + 50 v}{v}}$

Step 6.3

Factor $0.1$ out of $- 0.9 + 50 v$ .

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Step 6.3.1

Factor $0.1$ out of $- 0.9$ .

$y = \pm \sqrt{\frac{0.1 (- 9) + 50 v}{v}}$

Step 6.3.2

Factor $0.1$ out of $50 v$ .

$y = \pm \sqrt{\frac{0.1 (- 9) + 0.1 (500 v)}{v}}$

Step 6.3.3

Factor $0.1$ out of $0.1 (- 9) + 0.1 (500 v)$ .

$y = \pm \sqrt{\frac{0.1 (- 9 + 500 v)}{v}}$

Step 6.4

Rewrite $\sqrt{\frac{0.1 (- 9 + 500 v)}{v}}$ as $\frac{\sqrt{0.1 (- 9 + 500 v)}}{\sqrt{v}}$ .

$y = \pm \frac{\sqrt{0.1 (- 9 + 500 v)}}{\sqrt{v}}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Rewrite $0.1 (- 9 + 500 v)$ as ${({0.56234132}^{2})}^{2} (- 9 + 500 v)$ .

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Step 6.5.1.1

Rewrite $0.1$ as ${0.31622776}^{2}$ .

$y = \pm \frac{\sqrt{{0.31622776}^{2} (- 9 + 500 v)}}{\sqrt{v}}$

Step 6.5.1.2

Rewrite $0.31622776$ as ${0.56234132}^{2}$ .

$y = \pm \frac{\sqrt{{({0.56234132}^{2})}^{2} (- 9 + 500 v)}}{\sqrt{v}}$

Step 6.5.2

Pull terms out from under the radical.

$y = \pm \frac{{0.56234132}^{2} \sqrt{- 9 + 500 v}}{\sqrt{v}}$

Step 6.5.3

Raise $0.56234132$ to the power of $2$ .

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v}}{\sqrt{v}}$

Step 6.6

Multiply $\frac{0.31622776 \sqrt{- 9 + 500 v}}{\sqrt{v}}$ by $\frac{\sqrt{v}}{\sqrt{v}}$ .

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v}}{\sqrt{v}} \cdot \frac{\sqrt{v}}{\sqrt{v}}$

Step 6.7

Combine and simplify the denominator.

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Step 6.7.1

Multiply $\frac{0.31622776 \sqrt{- 9 + 500 v}}{\sqrt{v}}$ by $\frac{\sqrt{v}}{\sqrt{v}}$ .

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v} \sqrt{v}}{\sqrt{v} \sqrt{v}}$

Step 6.7.2

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

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v} \sqrt{v}}{{\sqrt{v}}^{1} \sqrt{v}}$

Step 6.7.3

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

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v} \sqrt{v}}{{\sqrt{v}}^{1} {\sqrt{v}}^{1}}$

Step 6.7.4

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

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v} \sqrt{v}}{{\sqrt{v}}^{1 + 1}}$

Step 6.7.5

Add $1$ and $1$ .

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v} \sqrt{v}}{{\sqrt{v}}^{2}}$

Step 6.7.6

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

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Step 6.7.6.1

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

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v} \sqrt{v}}{{(v^{\frac{1}{2}})}^{2}}$

Step 6.7.6.2

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

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v} \sqrt{v}}{v^{\frac{1}{2} \cdot 2}}$

Step 6.7.6.3

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

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v} \sqrt{v}}{v^{\frac{2}{2}}}$

Step 6.7.6.4

Cancel the common factor of $2$ .

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Step 6.7.6.4.1

Cancel the common factor.

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v} \sqrt{v}}{v^{\frac{2}{2}}}$

Step 6.7.6.4.2

Rewrite the expression.

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v} \sqrt{v}}{v^{1}}$

Step 6.7.6.5

Simplify.

$y = \pm \frac{0.31622776 \sqrt{- 9 + 500 v} \sqrt{v}}{v}$

Step 6.8

Combine using the product rule for radicals.

$y = \pm \frac{0.31622776 \sqrt{v (- 9 + 500 v)}}{v}$

Step 7

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

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Step 7.1

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

$y = \frac{0.31622776 \sqrt{v (- 9 + 500 v)}}{v}$

Step 7.2

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

$y = - \frac{0.31622776 \sqrt{v (- 9 + 500 v)}}{v}$

Step 7.3

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

$y = \frac{0.31622776 \sqrt{v (- 9 + 500 v)}}{v}$

$y = - \frac{0.31622776 \sqrt{v (- 9 + 500 v)}}{v}$

$y = \frac{0.31622776 \sqrt{v (- 9 + 500 v)}}{v}$

$y = - \frac{0.31622776 \sqrt{v (- 9 + 500 v)}}{v}$