Basic Math Examples

$30 y = 60 y^{3}$

Step 1

Subtract $60 y^{3}$ from both sides of the equation.

$30 y - 60 y^{3} = 0$

Step 2

Factor $30 y$ out of $30 y - 60 y^{3}$ .

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Step 2.1

Factor $30 y$ out of $30 y$ .

$30 y (1) - 60 y^{3} = 0$

Step 2.2

Factor $30 y$ out of $- 60 y^{3}$ .

$30 y (1) + 30 y (- 2 y^{2}) = 0$

Step 2.3

Factor $30 y$ out of $30 y (1) + 30 y (- 2 y^{2})$ .

$30 y (1 - 2 y^{2}) = 0$

Step 3

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

$y = 0$

$1 - 2 y^{2} = 0$

Step 4

Set $y$ equal to $0$ .

$y = 0$

Step 5

Set $1 - 2 y^{2}$ equal to $0$ and solve for $y$ .

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Step 5.1

Set $1 - 2 y^{2}$ equal to $0$ .

$1 - 2 y^{2} = 0$

Step 5.2

Solve $1 - 2 y^{2} = 0$ for $y$ .

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Step 5.2.1

Subtract $1$ from both sides of the equation.

$- 2 y^{2} = - 1$

Step 5.2.2

Divide each term in $- 2 y^{2} = - 1$ by $- 2$ and simplify.

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Step 5.2.2.1

Divide each term in $- 2 y^{2} = - 1$ by $- 2$ .

$\frac{- 2 y^{2}}{- 2} = \frac{- 1}{- 2}$

Step 5.2.2.2

Simplify the left side.

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Step 5.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 y^{2}}{- 2} = \frac{- 1}{- 2}$

Step 5.2.2.2.1.2

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

$y^{2} = \frac{- 1}{- 2}$

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.

$y^{2} = \frac{1}{2}$

Step 5.2.3

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

$y = \pm \sqrt{\frac{1}{2}}$

Step 5.2.4

Simplify $\pm \sqrt{\frac{1}{2}}$ .

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Step 5.2.4.1

Rewrite $\sqrt{\frac{1}{2}}$ as $\frac{\sqrt{1}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{1}}{\sqrt{2}}$

Step 5.2.4.2

Any root of $1$ is $1$ .

$y = \pm \frac{1}{\sqrt{2}}$

Step 5.2.4.3

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

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

Step 5.2.4.4

Combine and simplify the denominator.

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Step 5.2.4.4.1

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$y = \pm \frac{\sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 5.2.4.4.2

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.2.4.4.3

Raise $\sqrt{2}$ to the power of $1$ .

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

Step 5.2.4.4.4

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

$y = \pm \frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 5.2.4.4.5

Add $1$ and $1$ .

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

Step 5.2.4.4.6

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

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Step 5.2.4.4.6.1

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

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

Step 5.2.4.4.6.2

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

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

Step 5.2.4.4.6.3

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

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

Step 5.2.4.4.6.4

Cancel the common factor of $2$ .

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Step 5.2.4.4.6.4.1

Cancel the common factor.

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

Step 5.2.4.4.6.4.2

Rewrite the expression.

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

Step 5.2.4.4.6.5

Evaluate the exponent.

$y = \pm \frac{\sqrt{2}}{2}$

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.

$y = \frac{\sqrt{2}}{2}$

Step 5.2.5.2

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

$y = - \frac{\sqrt{2}}{2}$

Step 5.2.5.3

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

$y = \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 6

The final solution is all the values that make $30 y (1 - 2 y^{2}) = 0$ true.

$y = 0, \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$y = 0, \frac{\sqrt{2}}{2}, - \frac{\sqrt{2}}{2}$

Decimal Form:

$y = 0, 0.70710678 \dots, - 0.70710678 \dots$