Calculus Examples

$2.8 y = y^{3}$

Step 1

Subtract $y^{3}$ from both sides of the equation.

$2.8 y - y^{3} = 0$

Step 2

Factor $0.2 y$ out of $2.8 y - y^{3}$ .

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Step 2.1

Factor $0.2 y$ out of $2.8 y$ .

$0.2 y (14) - y^{3} = 0$

Step 2.2

Factor $0.2 y$ out of $- y^{3}$ .

$0.2 y (14) + 0.2 y (- 5 y^{2}) = 0$

Step 2.3

Factor $0.2 y$ out of $0.2 y (14) + 0.2 y (- 5 y^{2})$ .

$0.2 y (14 - 5 y^{2}) = 0$

Step 3

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

$y = 0$

$14 - 5 y^{2} = 0$

Step 4

Set $y$ equal to $0$ .

$y = 0$

Step 5

Set $14 - 5 y^{2}$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $14 - 5 y^{2}$ equal to $0$ .

$14 - 5 y^{2} = 0$

Step 5.2

Solve $14 - 5 y^{2} = 0$ for $y$ .

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Step 5.2.1

Subtract $14$ from both sides of the equation.

$- 5 y^{2} = - 14$

Step 5.2.2

Divide each term in $- 5 y^{2} = - 14$ by $- 5$ and simplify.

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Step 5.2.2.1

Divide each term in $- 5 y^{2} = - 14$ by $- 5$ .

$\frac{- 5 y^{2}}{- 5} = \frac{- 14}{- 5}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $- 5$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{- 5 y^{2}}{- 5} = \frac{- 14}{- 5}$

Step 5.2.2.2.1.2

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

$y^{2} = \frac{- 14}{- 5}$

Step 5.2.2.3

Simplify the right side.

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Step 5.2.2.3.1

Dividing two negative values results in a positive value.

$y^{2} = \frac{14}{5}$

Step 5.2.3

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

$y = \pm \sqrt{\frac{14}{5}}$

Step 5.2.4

Simplify $\pm \sqrt{\frac{14}{5}}$ .

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Step 5.2.4.1

Rewrite $\sqrt{\frac{14}{5}}$ as $\frac{\sqrt{14}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{14}}{\sqrt{5}}$

Step 5.2.4.2

Multiply $\frac{\sqrt{14}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{14}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}$

Step 5.2.4.3

Combine and simplify the denominator.

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Step 5.2.4.3.1

Multiply $\frac{\sqrt{14}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$y = \pm \frac{\sqrt{14} \sqrt{5}}{\sqrt{5} \sqrt{5}}$

Step 5.2.4.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$y = \pm \frac{\sqrt{14} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}}$

Step 5.2.4.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$y = \pm \frac{\sqrt{14} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}}$

Step 5.2.4.3.4

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

$y = \pm \frac{\sqrt{14} \sqrt{5}}{{\sqrt{5}}^{1 + 1}}$

Step 5.2.4.3.5

Add $1$ and $1$ .

$y = \pm \frac{\sqrt{14} \sqrt{5}}{{\sqrt{5}}^{2}}$

Step 5.2.4.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.2.4.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$y = \pm \frac{\sqrt{14} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}}$

Step 5.2.4.3.6.2

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

$y = \pm \frac{\sqrt{14} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}}$

Step 5.2.4.3.6.3

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

$y = \pm \frac{\sqrt{14} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.2.4.3.6.4

Cancel the common factor of $2$ .

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Step 5.2.4.3.6.4.1

Cancel the common factor.

$y = \pm \frac{\sqrt{14} \sqrt{5}}{5^{\frac{2}{2}}}$

Step 5.2.4.3.6.4.2

Rewrite the expression.

$y = \pm \frac{\sqrt{14} \sqrt{5}}{5^{1}}$

Step 5.2.4.3.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{14} \sqrt{5}}{5}$

Step 5.2.4.4

Simplify the numerator.

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Step 5.2.4.4.1

Combine using the product rule for radicals.

$y = \pm \frac{\sqrt{14 \cdot 5}}{5}$

Step 5.2.4.4.2

Multiply $14$ by $5$ .

$y = \pm \frac{\sqrt{70}}{5}$

Step 5.2.5

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

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Step 5.2.5.1

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

$y = \frac{\sqrt{70}}{5}$

Step 5.2.5.2

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

$y = - \frac{\sqrt{70}}{5}$

Step 5.2.5.3

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

$y = \frac{\sqrt{70}}{5}, - \frac{\sqrt{70}}{5}$

Step 6

The final solution is all the values that make $0.2 y (14 - 5 y^{2}) = 0$ true.

$y = 0, \frac{\sqrt{70}}{5}, - \frac{\sqrt{70}}{5}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = 0, \frac{\sqrt{70}}{5}, - \frac{\sqrt{70}}{5}$

Decimal Form:

$y = 0, 1.67332005 \dots, - 1.67332005 \dots$