Precalculus Examples

$y = x - \frac{1}{75} x^{3}$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = x - \frac{1}{75} x^{3}$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $x - \frac{1}{75} x^{3} = 0$ .

$x - \frac{1}{75} x^{3} = 0$

Step 1.2.2

Combine $x^{3}$ and $\frac{1}{75}$ .

$x - \frac{x^{3}}{75} = 0$

Step 1.2.3

Factor $x$ out of $x - \frac{x^{3}}{75}$ .

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Step 1.2.3.1

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

$x \cdot 1 - \frac{x^{3}}{75} = 0$

Step 1.2.3.2

Factor $x$ out of $- \frac{x^{3}}{75}$ .

$x \cdot 1 + x (- \frac{x^{2}}{75}) = 0$

Step 1.2.3.3

Factor $x$ out of $x \cdot 1 + x (- \frac{x^{2}}{75})$ .

$x (1 - \frac{x^{2}}{75}) = 0$

Step 1.2.4

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

$x = 0$

$1 - \frac{x^{2}}{75} = 0$

Step 1.2.5

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.6

Set $1 - \frac{x^{2}}{75}$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $1 - \frac{x^{2}}{75}$ equal to $0$ .

$1 - \frac{x^{2}}{75} = 0$

Step 1.2.6.2

Solve $1 - \frac{x^{2}}{75} = 0$ for $x$ .

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Step 1.2.6.2.1

Subtract $1$ from both sides of the equation.

$- \frac{x^{2}}{75} = - 1$

Step 1.2.6.2.2

Multiply both sides of the equation by $- 75$ .

$- 75 (- \frac{x^{2}}{75}) = - 75 \cdot - 1$

Step 1.2.6.2.3

Simplify both sides of the equation.

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Step 1.2.6.2.3.1

Simplify the left side.

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Step 1.2.6.2.3.1.1

Simplify $- 75 (- \frac{x^{2}}{75})$ .

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Step 1.2.6.2.3.1.1.1

Cancel the common factor of $75$ .

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Step 1.2.6.2.3.1.1.1.1

Move the leading negative in $- \frac{x^{2}}{75}$ into the numerator.

$- 75 \frac{- x^{2}}{75} = - 75 \cdot - 1$

Step 1.2.6.2.3.1.1.1.2

Factor $75$ out of $- 75$ .

$75 (- 1) \frac{- x^{2}}{75} = - 75 \cdot - 1$

Step 1.2.6.2.3.1.1.1.3

Cancel the common factor.

$75 \cdot - 1 \frac{- x^{2}}{75} = - 75 \cdot - 1$

Step 1.2.6.2.3.1.1.1.4

Rewrite the expression.

$- - x^{2} = - 75 \cdot - 1$

Step 1.2.6.2.3.1.1.2

Multiply.

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Step 1.2.6.2.3.1.1.2.1

Multiply $- 1$ by $- 1$ .

$1 x^{2} = - 75 \cdot - 1$

Step 1.2.6.2.3.1.1.2.2

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

$x^{2} = - 75 \cdot - 1$

Step 1.2.6.2.3.2

Simplify the right side.

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Step 1.2.6.2.3.2.1

Multiply $- 75$ by $- 1$ .

$x^{2} = 75$

Step 1.2.6.2.4

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

$x = \pm \sqrt{75}$

Step 1.2.6.2.5

Simplify $\pm \sqrt{75}$ .

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Step 1.2.6.2.5.1

Rewrite $75$ as $5^{2} \cdot 3$ .

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Step 1.2.6.2.5.1.1

Factor $25$ out of $75$ .

$x = \pm \sqrt{25 (3)}$

Step 1.2.6.2.5.1.2

Rewrite $25$ as $5^{2}$ .

$x = \pm \sqrt{5^{2} \cdot 3}$

Step 1.2.6.2.5.2

Pull terms out from under the radical.

$x = \pm 5 \sqrt{3}$

Step 1.2.6.2.6

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

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Step 1.2.6.2.6.1

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

$x = 5 \sqrt{3}$

Step 1.2.6.2.6.2

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

$x = - 5 \sqrt{3}$

Step 1.2.6.2.6.3

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

$x = 5 \sqrt{3}, - 5 \sqrt{3}$

Step 1.2.7

The final solution is all the values that make $x (1 - \frac{x^{2}}{75}) = 0$ true.

$x = 0, 5 \sqrt{3}, - 5 \sqrt{3}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (5 \sqrt{3}, 0), (- 5 \sqrt{3}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = (0) - \frac{1}{75} \cdot {(0)}^{3}$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 0 - \frac{1}{75} \cdot 0^{3}$

Step 2.2.2

Remove parentheses.

$y = (0) - \frac{1}{75} \cdot {(0)}^{3}$

Step 2.2.3

Simplify $(0) - \frac{1}{75} \cdot {(0)}^{3}$ .

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Raising $0$ to any positive power yields $0$ .

$y = 0 - \frac{1}{75} \cdot 0$

Step 2.2.3.1.2

Multiply $- \frac{1}{75} \cdot 0$ .

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Step 2.2.3.1.2.1

Multiply $0$ by $- 1$ .

$y = 0 + 0 (\frac{1}{75})$

Step 2.2.3.1.2.2

Multiply $0$ by $\frac{1}{75}$ .

$y = 0 + 0$

Step 2.2.3.2

Add $0$ and $0$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 0)$

Step 3

List the intersections.

x-intercept(s): $(0, 0), (5 \sqrt{3}, 0), (- 5 \sqrt{3}, 0)$

y-intercept(s): $(0, 0)$

Step 4