Calculus Examples

$0 = x - \frac{1}{60} x^{3}$

Step 1

Rewrite the equation as $x - \frac{1}{60} x^{3} = 0$ .

$x - \frac{1}{60} x^{3} = 0$

Step 2

Combine $x^{3}$ and $\frac{1}{60}$ .

$x - \frac{x^{3}}{60} = 0$

Step 3

Factor $x$ out of $x - \frac{x^{3}}{60}$ .

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Step 3.1

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - \frac{x^{3}}{60} = 0$

Step 3.2

Factor $x$ out of $- \frac{x^{3}}{60}$ .

$x \cdot 1 + x (- \frac{x^{2}}{60}) = 0$

Step 3.3

Factor $x$ out of $x \cdot 1 + x (- \frac{x^{2}}{60})$ .

$x (1 - \frac{x^{2}}{60}) = 0$

Step 4

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

$x = 0$

$1 - \frac{x^{2}}{60} = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $1 - \frac{x^{2}}{60}$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $1 - \frac{x^{2}}{60}$ equal to $0$ .

$1 - \frac{x^{2}}{60} = 0$

Step 6.2

Solve $1 - \frac{x^{2}}{60} = 0$ for $x$ .

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Step 6.2.1

Subtract $1$ from both sides of the equation.

$- \frac{x^{2}}{60} = - 1$

Step 6.2.2

Multiply both sides of the equation by $- 60$ .

$- 60 (- \frac{x^{2}}{60}) = - 60 \cdot - 1$

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 $- 60 (- \frac{x^{2}}{60})$ .

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Step 6.2.3.1.1.1

Cancel the common factor of $60$ .

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Step 6.2.3.1.1.1.1

Move the leading negative in $- \frac{x^{2}}{60}$ into the numerator.

$- 60 \frac{- x^{2}}{60} = - 60 \cdot - 1$

Step 6.2.3.1.1.1.2

Factor $60$ out of $- 60$ .

$60 (- 1) \frac{- x^{2}}{60} = - 60 \cdot - 1$

Step 6.2.3.1.1.1.3

Cancel the common factor.

$60 \cdot - 1 \frac{- x^{2}}{60} = - 60 \cdot - 1$

Step 6.2.3.1.1.1.4

Rewrite the expression.

$- - x^{2} = - 60 \cdot - 1$

Step 6.2.3.1.1.2

Multiply.

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Step 6.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - 60 \cdot - 1$

Step 6.2.3.1.1.2.2

Multiply $x^{2}$ by $1$ .

$x^{2} = - 60 \cdot - 1$

Step 6.2.3.2

Simplify the right side.

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Step 6.2.3.2.1

Multiply $- 60$ by $- 1$ .

$x^{2} = 60$

Step 6.2.4

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

$x = \pm \sqrt{60}$

Step 6.2.5

Simplify $\pm \sqrt{60}$ .

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Step 6.2.5.1

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 6.2.5.1.1

Factor $4$ out of $60$ .

$x = \pm \sqrt{4 (15)}$

Step 6.2.5.1.2

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2} \cdot 15}$

Step 6.2.5.2

Pull terms out from under the radical.

$x = \pm 2 \sqrt{15}$

Step 6.2.6

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

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Step 6.2.6.1

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

$x = 2 \sqrt{15}$

Step 6.2.6.2

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

$x = - 2 \sqrt{15}$

Step 6.2.6.3

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

$x = 2 \sqrt{15}, - 2 \sqrt{15}$

Step 7

The final solution is all the values that make $x (1 - \frac{x^{2}}{60}) = 0$ true.

$x = 0, 2 \sqrt{15}, - 2 \sqrt{15}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = 0, 2 \sqrt{15}, - 2 \sqrt{15}$

Decimal Form:

$x = 0, 7.74596669 \dots, - 7.74596669 \dots$