Calculus Examples

$x + \frac{108}{x^{2}}$

Step 1

Write $x + \frac{108}{x^{2}}$ as an equation.

$y = x + \frac{108}{x^{2}}$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = x + \frac{108}{x^{2}}$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $x + \frac{108}{x^{2}} = 0$ .

$x + \frac{108}{x^{2}} = 0$

Step 2.2.2

Find the LCD of the terms in the equation.

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Step 2.2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, x^{2}, 1$

Step 2.2.2.2

The LCM of one and any expression is the expression.

$x^{2}$

Step 2.2.3

Multiply each term in $x + \frac{108}{x^{2}} = 0$ by $x^{2}$ to eliminate the fractions.

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Step 2.2.3.1

Multiply each term in $x + \frac{108}{x^{2}} = 0$ by $x^{2}$ .

$x \cdot x^{2} + \frac{108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Simplify each term.

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Step 2.2.3.2.1.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 2.2.3.2.1.1.1

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

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Step 2.2.3.2.1.1.1.1

Raise $x$ to the power of $1$ .

$x^{1} x^{2} + \frac{108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.3.2.1.1.1.2

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

$x^{1 + 2} + \frac{108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.3.2.1.1.2

Add $1$ and $2$ .

$x^{3} + \frac{108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.3.2.1.2

Cancel the common factor of $x^{2}$ .

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Step 2.2.3.2.1.2.1

Cancel the common factor.

$x^{3} + \frac{108}{x^{2}} x^{2} = 0 x^{2}$

Step 2.2.3.2.1.2.2

Rewrite the expression.

$x^{3} + 108 = 0 x^{2}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Multiply $0$ by $x^{2}$ .

$x^{3} + 108 = 0$

Step 2.2.4

Solve the equation.

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Step 2.2.4.1

Subtract $108$ from both sides of the equation.

$x^{3} = - 108$

Step 2.2.4.2

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

$x = \sqrt[3]{- 108}$

Step 2.2.4.3

Simplify $\sqrt[3]{- 108}$ .

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Step 2.2.4.3.1

Rewrite $- 108$ as ${(- 3)}^{3} \cdot 4$ .

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Step 2.2.4.3.1.1

Factor $- 27$ out of $- 108$ .

$x = \sqrt[3]{- 27 (4)}$

Step 2.2.4.3.1.2

Rewrite $- 27$ as ${(- 3)}^{3}$ .

$x = \sqrt[3]{{(- 3)}^{3} \cdot 4}$

Step 2.2.4.3.2

Pull terms out from under the radical.

$x = - 3 \sqrt[3]{4}$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(- 3 \sqrt[3]{4}, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = (0) + \frac{108}{{(0)}^{2}}$

Step 3.2

The equation has an undefined fraction.

Undefined

Step 3.3

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

y-intercept(s): $None$

Step 4

List the intersections.

x-intercept(s): $(- 3 \sqrt[3]{4}, 0)$

y-intercept(s): $None$

Step 5