Calculus Examples

$4 - \frac{108}{x^{3}} = 0$

Step 1

Subtract $4$ from both sides of the equation.

$- \frac{108}{x^{3}} = - 4$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$x^{3}, 1$

Step 2.2

The LCM of one and any expression is the expression.

$x^{3}$

Step 3

Multiply each term in $- \frac{108}{x^{3}} = - 4$ by $x^{3}$ to eliminate the fractions.

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Step 3.1

Multiply each term in $- \frac{108}{x^{3}} = - 4$ by $x^{3}$ .

$- \frac{108}{x^{3}} x^{3} = - 4 x^{3}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x^{3}$ .

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Step 3.2.1.1

Move the leading negative in $- \frac{108}{x^{3}}$ into the numerator.

$\frac{- 108}{x^{3}} x^{3} = - 4 x^{3}$

Step 3.2.1.2

Cancel the common factor.

$\frac{- 108}{x^{3}} x^{3} = - 4 x^{3}$

Step 3.2.1.3

Rewrite the expression.

$- 108 = - 4 x^{3}$

Step 4

Solve the equation.

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Step 4.1

Rewrite the equation as $- 4 x^{3} = - 108$ .

$- 4 x^{3} = - 108$

Step 4.2

Add $108$ to both sides of the equation.

$- 4 x^{3} + 108 = 0$

Step 4.3

Factor the left side of the equation.

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Step 4.3.1

Factor $- 4$ out of $- 4 x^{3} + 108$ .

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Step 4.3.1.1

Factor $- 4$ out of $- 4 x^{3}$ .

$- 4 x^{3} + 108 = 0$

Step 4.3.1.2

Factor $- 4$ out of $108$ .

$- 4 x^{3} - 4 \cdot - 27 = 0$

Step 4.3.1.3

Factor $- 4$ out of $- 4 (x^{3}) - 4 (- 27)$ .

$- 4 (x^{3} - 27) = 0$

Step 4.3.2

Rewrite $27$ as $3^{3}$ .

$- 4 (x^{3} - 3^{3}) = 0$

Step 4.3.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 3$ .

$- 4 ((x - 3) (x^{2} + x \cdot 3 + 3^{2})) = 0$

Step 4.3.4

Factor.

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Step 4.3.4.1

Simplify.

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Step 4.3.4.1.1

Move $3$ to the left of $x$ .

$- 4 ((x - 3) (x^{2} + 3 x + 3^{2})) = 0$

Step 4.3.4.1.2

Raise $3$ to the power of $2$ .

$- 4 ((x - 3) (x^{2} + 3 x + 9)) = 0$

Step 4.3.4.2

Remove unnecessary parentheses.

$- 4 (x - 3) (x^{2} + 3 x + 9) = 0$

Step 4.4

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

$x - 3 = 0$

$x^{2} + 3 x + 9 = 0$

Step 4.5

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.5.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.5.2

Add $3$ to both sides of the equation.

$x = 3$

Step 4.6

Set $x^{2} + 3 x + 9$ equal to $0$ and solve for $x$ .

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Step 4.6.1

Set $x^{2} + 3 x + 9$ equal to $0$ .

$x^{2} + 3 x + 9 = 0$

Step 4.6.2

Solve $x^{2} + 3 x + 9 = 0$ for $x$ .

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Step 4.6.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 4.6.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 9$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 9)}}{2 \cdot 1}$

Step 4.6.2.3

Simplify.

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Step 4.6.2.3.1

Simplify the numerator.

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Step 4.6.2.3.1.1

Raise $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}$

Step 4.6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 9$ .

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Step 4.6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 9}}{2 \cdot 1}$

Step 4.6.2.3.1.2.2

Multiply $- 4$ by $9$ .

$x = \frac{- 3 \pm \sqrt{9 - 36}}{2 \cdot 1}$

Step 4.6.2.3.1.3

Subtract $36$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 27}}{2 \cdot 1}$

Step 4.6.2.3.1.4

Rewrite $- 27$ as $- 1 (27)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 27}}{2 \cdot 1}$

Step 4.6.2.3.1.5

Rewrite $\sqrt{- 1 (27)}$ as $\sqrt{- 1} \cdot \sqrt{27}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{27}}{2 \cdot 1}$

Step 4.6.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \cdot \sqrt{27}}{2 \cdot 1}$

Step 4.6.2.3.1.7

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 4.6.2.3.1.7.1

Factor $9$ out of $27$ .

$x = \frac{- 3 \pm i \cdot \sqrt{9 (3)}}{2 \cdot 1}$

Step 4.6.2.3.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{- 3 \pm i \cdot \sqrt{3^{2} \cdot 3}}{2 \cdot 1}$

Step 4.6.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 3 \pm i \cdot (3 \sqrt{3})}{2 \cdot 1}$

Step 4.6.2.3.1.9

Move $3$ to the left of $i$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2 \cdot 1}$

Step 4.6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm 3 i \sqrt{3}}{2}$

Step 4.6.2.4

The final answer is the combination of both solutions.

$x = - \frac{3 - 3 i \sqrt{3}}{2}, - \frac{3 + 3 i \sqrt{3}}{2}$

Step 4.7

The final solution is all the values that make $- 4 (x - 3) (x^{2} + 3 x + 9) = 0$ true.

$x = 3, - \frac{3 - 3 i \sqrt{3}}{2}, - \frac{3 + 3 i \sqrt{3}}{2}$