Pre-Algebra Examples

$x^{3} = \frac{125}{216}$

Step 1

Subtract $\frac{125}{216}$ from both sides of the equation.

$x^{3} - \frac{125}{216} = 0$

Step 2

Factor the left side of the equation.

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Step 2.1

Rewrite $\frac{125}{216}$ as ${(\frac{5}{6})}^{3}$ .

$x^{3} - {(\frac{5}{6})}^{3} = 0$

Step 2.2

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 $b = \frac{5}{6}$ .

$(x - \frac{5}{6}) (x^{2} + x (\frac{5}{6}) + {(\frac{5}{6})}^{2}) = 0$

Step 2.3

Simplify.

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Step 2.3.1

Combine $x$ and $\frac{5}{6}$ .

$(x - \frac{5}{6}) (x^{2} + \frac{x \cdot 5}{6} + {(\frac{5}{6})}^{2}) = 0$

Step 2.3.2

Move $5$ to the left of $x$ .

$(x - \frac{5}{6}) (x^{2} + \frac{5 \cdot x}{6} + {(\frac{5}{6})}^{2}) = 0$

Step 2.3.3

Multiply $5$ by $x$ .

$(x - \frac{5}{6}) (x^{2} + \frac{5 x}{6} + {(\frac{5}{6})}^{2}) = 0$

Step 2.3.4

Apply the product rule to $\frac{5}{6}$ .

$(x - \frac{5}{6}) (x^{2} + \frac{5 x}{6} + \frac{5^{2}}{6^{2}}) = 0$

Step 2.3.5

Raise $5$ to the power of $2$ .

$(x - \frac{5}{6}) (x^{2} + \frac{5 x}{6} + \frac{25}{6^{2}}) = 0$

Step 2.3.6

Raise $6$ to the power of $2$ .

$(x - \frac{5}{6}) (x^{2} + \frac{5 x}{6} + \frac{25}{36}) = 0$

Step 3

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

$x - \frac{5}{6} = 0$

$x^{2} + \frac{5 x}{6} + \frac{25}{36} = 0$

Step 4

Set $x - \frac{5}{6}$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - \frac{5}{6}$ equal to $0$ .

$x - \frac{5}{6} = 0$

Step 4.2

Add $\frac{5}{6}$ to both sides of the equation.

$x = \frac{5}{6}$

Step 5

Set $x^{2} + \frac{5 x}{6} + \frac{25}{36}$ equal to $0$ and solve for $x$ .

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Step 5.1

Set $x^{2} + \frac{5 x}{6} + \frac{25}{36}$ equal to $0$ .

$x^{2} + \frac{5 x}{6} + \frac{25}{36} = 0$

Step 5.2

Solve $x^{2} + \frac{5 x}{6} + \frac{25}{36} = 0$ for $x$ .

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Step 5.2.1

Multiply through by the least common denominator $36$ , then simplify.

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Step 5.2.1.1

Apply the distributive property.

$36 x^{2} + 36 (\frac{5 x}{6}) + 36 (\frac{25}{36}) = 0$

Step 5.2.1.2

Simplify.

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Step 5.2.1.2.1

Cancel the common factor of $6$ .

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Step 5.2.1.2.1.1

Factor $6$ out of $36$ .

$36 x^{2} + 6 (6) (\frac{5 x}{6}) + 36 (\frac{25}{36}) = 0$

Step 5.2.1.2.1.2

Cancel the common factor.

$36 x^{2} + 6 \cdot (6 (\frac{5 x}{6})) + 36 (\frac{25}{36}) = 0$

Step 5.2.1.2.1.3

Rewrite the expression.

$36 x^{2} + 6 (5 x) + 36 (\frac{25}{36}) = 0$

Step 5.2.1.2.2

Multiply $5$ by $6$ .

$36 x^{2} + 30 x + 36 (\frac{25}{36}) = 0$

Step 5.2.1.2.3

Cancel the common factor of $36$ .

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Step 5.2.1.2.3.1

Cancel the common factor.

$36 x^{2} + 30 x + 36 (\frac{25}{36}) = 0$

Step 5.2.1.2.3.2

Rewrite the expression.

$36 x^{2} + 30 x + 25 = 0$

Step 5.2.2

Use the quadratic formula to find the solutions.

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

Step 5.2.3

Substitute the values $a = 36$ , $b = 30$ , and $c = 25$ into the quadratic formula and solve for $x$ .

$\frac{- 30 \pm \sqrt{30^{2} - 4 \cdot (36 \cdot 25)}}{2 \cdot 36}$

Step 5.2.4

Simplify.

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Step 5.2.4.1

Simplify the numerator.

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Step 5.2.4.1.1

Raise $30$ to the power of $2$ .

$x = \frac{- 30 \pm \sqrt{900 - 4 \cdot 36 \cdot 25}}{2 \cdot 36}$

Step 5.2.4.1.2

Multiply $- 4 \cdot 36 \cdot 25$ .

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Step 5.2.4.1.2.1

Multiply $- 4$ by $36$ .

$x = \frac{- 30 \pm \sqrt{900 - 144 \cdot 25}}{2 \cdot 36}$

Step 5.2.4.1.2.2

Multiply $- 144$ by $25$ .

$x = \frac{- 30 \pm \sqrt{900 - 3600}}{2 \cdot 36}$

Step 5.2.4.1.3

Subtract $3600$ from $900$ .

$x = \frac{- 30 \pm \sqrt{- 2700}}{2 \cdot 36}$

Step 5.2.4.1.4

Rewrite $- 2700$ as $- 1 (2700)$ .

$x = \frac{- 30 \pm \sqrt{- 1 \cdot 2700}}{2 \cdot 36}$

Step 5.2.4.1.5

Rewrite $\sqrt{- 1 (2700)}$ as $\sqrt{- 1} \cdot \sqrt{2700}$ .

$x = \frac{- 30 \pm \sqrt{- 1} \cdot \sqrt{2700}}{2 \cdot 36}$

Step 5.2.4.1.6

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

$x = \frac{- 30 \pm i \cdot \sqrt{2700}}{2 \cdot 36}$

Step 5.2.4.1.7

Rewrite $2700$ as $30^{2} \cdot 3$ .

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Step 5.2.4.1.7.1

Factor $900$ out of $2700$ .

$x = \frac{- 30 \pm i \cdot \sqrt{900 (3)}}{2 \cdot 36}$

Step 5.2.4.1.7.2

Rewrite $900$ as $30^{2}$ .

$x = \frac{- 30 \pm i \cdot \sqrt{30^{2} \cdot 3}}{2 \cdot 36}$

Step 5.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 30 \pm i \cdot (30 \sqrt{3})}{2 \cdot 36}$

Step 5.2.4.1.9

Move $30$ to the left of $i$ .

$x = \frac{- 30 \pm 30 i \sqrt{3}}{2 \cdot 36}$

Step 5.2.4.2

Multiply $2$ by $36$ .

$x = \frac{- 30 \pm 30 i \sqrt{3}}{72}$

Step 5.2.4.3

Simplify $\frac{- 30 \pm 30 i \sqrt{3}}{72}$ .

$x = \frac{- 5 \pm 5 i \sqrt{3}}{12}$

Step 5.2.5

The final answer is the combination of both solutions.

$x = - \frac{5 - 5 i \sqrt{3}}{12}, - \frac{5 + 5 i \sqrt{3}}{12}$

Step 6

The final solution is all the values that make $(x - \frac{5}{6}) (x^{2} + \frac{5 x}{6} + \frac{25}{36}) = 0$ true.

$x = \frac{5}{6}, - \frac{5 - 5 i \sqrt{3}}{12}, - \frac{5 + 5 i \sqrt{3}}{12}$