Finite Math Examples

$\frac{1}{4} \cdot {| x^{3} + 1 |}^{2} = | x^{3} + 1 | - 1$

Step 1

Simplify $\frac{1}{4} {| x^{3} + 1 |}^{2}$ .

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Step 1.1

Rewrite.

$0 + 0 + \frac{1}{4} {| x^{3} + 1 |}^{2} = | x^{3} + 1 | - 1$

Step 1.2

Simplify by adding zeros.

$\frac{1}{4} {| x^{3} + 1 |}^{2} = | x^{3} + 1 | - 1$

Step 1.3

Combine $\frac{1}{4}$ and ${| x^{3} + 1 |}^{2}$ .

$\frac{{| x^{3} + 1 |}^{2}}{4} = | x^{3} + 1 | - 1$

Step 2

Subtract $| x^{3} + 1 |$ from both sides of the equation.

$\frac{{| x^{3} + 1 |}^{2}}{4} - | x^{3} + 1 | = - 1$

Step 3

Multiply each term in $\frac{{| x^{3} + 1 |}^{2}}{4} - | x^{3} + 1 | = - 1$ by $4$ to eliminate the fractions.

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Step 3.1

Multiply each term in $\frac{{| x^{3} + 1 |}^{2}}{4} - | x^{3} + 1 | = - 1$ by $4$ .

$\frac{{| x^{3} + 1 |}^{2}}{4} \cdot 4 - | x^{3} + 1 | \cdot 4 = - 1 \cdot 4$

Step 3.2

Simplify the left side.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1

Cancel the common factor.

$\frac{{| x^{3} + 1 |}^{2}}{4} \cdot 4 - | x^{3} + 1 | \cdot 4 = - 1 \cdot 4$

Step 3.2.1.1.2

Rewrite the expression.

${| x^{3} + 1 |}^{2} - | x^{3} + 1 | \cdot 4 = - 1 \cdot 4$

Step 3.2.1.2

Multiply $4$ by $- 1$ .

${| x^{3} + 1 |}^{2} - 4 | x^{3} + 1 | = - 1 \cdot 4$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply $- 1$ by $4$ .

${| x^{3} + 1 |}^{2} - 4 | x^{3} + 1 | = - 4$

Step 4

Add $4$ to both sides of the equation.

${| x^{3} + 1 |}^{2} - 4 | x^{3} + 1 | + 4 = 0$

Step 5

Factor using the perfect square rule.

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Step 5.1

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

${| x^{3} + 1 |}^{2} - 4 | x^{3} + 1 | + 2^{2} = 0$

Step 5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 | x^{3} + 1 | = 2 \cdot | x^{3} + 1 | \cdot 2$

Step 5.3

Rewrite the polynomial.

${| x^{3} + 1 |}^{2} - 2 \cdot | x^{3} + 1 | \cdot 2 + 2^{2} = 0$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = | x^{3} + 1 |$ and $b = 2$ .

${(| x^{3} + 1 | - 2)}^{2} = 0$

Step 6

Set the $| x^{3} + 1 | - 2$ equal to $0$ .

$| x^{3} + 1 | - 2 = 0$

Step 7

Add $2$ to both sides of the equation.

$| x^{3} + 1 | = 2$

Step 8

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x^{3} + 1 = \pm 2$

Step 9

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

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Step 9.1

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

$x^{3} + 1 = 2$

Step 9.2

Move all terms not containing $x$ to the right side of the equation.

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Step 9.2.1

Subtract $1$ from both sides of the equation.

$x^{3} = 2 - 1$

Step 9.2.2

Subtract $1$ from $2$ .

$x^{3} = 1$

Step 9.3

Subtract $1$ from both sides of the equation.

$x^{3} - 1 = 0$

Step 9.4

Factor the left side of the equation.

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Step 9.4.1

Rewrite $1$ as $1^{3}$ .

$x^{3} - 1^{3} = 0$

Step 9.4.2

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 $b = 1$ .

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

Step 9.4.3

Simplify.

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Step 9.4.3.1

Multiply $x$ by $1$ .

$(x - 1) (x^{2} + x + 1^{2}) = 0$

Step 9.4.3.2

One to any power is one.

$(x - 1) (x^{2} + x + 1) = 0$

Step 9.5

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

$x - 1 = 0$

$x^{2} + x + 1 = 0$

Step 9.6

Set $x - 1$ equal to $0$ and solve for $x$ .

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Step 9.6.1

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

Step 9.6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 9.7

Set $x^{2} + x + 1$ equal to $0$ and solve for $x$ .

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Step 9.7.1

Set $x^{2} + x + 1$ equal to $0$ .

$x^{2} + x + 1 = 0$

Step 9.7.2

Solve $x^{2} + x + 1 = 0$ for $x$ .

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Step 9.7.2.1

Use the quadratic formula to find the solutions.

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

Step 9.7.2.2

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

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

Step 9.7.2.3

Simplify.

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Step 9.7.2.3.1

Simplify the numerator.

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Step 9.7.2.3.1.1

One to any power is one.

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

Step 9.7.2.3.1.2

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

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Step 9.7.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 9.7.2.3.1.2.2

Multiply $- 4$ by $1$ .

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

Step 9.7.2.3.1.3

Subtract $4$ from $1$ .

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

Step 9.7.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

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

Step 9.7.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

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

Step 9.7.2.3.1.6

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

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

Step 9.7.2.3.2

Multiply $2$ by $1$ .

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

Step 9.7.2.4

The final answer is the combination of both solutions.

$x = - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 9.8

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

$x = 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}$

Step 9.9

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

$x^{3} + 1 = - 2$

Step 9.10

Move all terms not containing $x$ to the right side of the equation.

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Step 9.10.1

Subtract $1$ from both sides of the equation.

$x^{3} = - 2 - 1$

Step 9.10.2

Subtract $1$ from $- 2$ .

$x^{3} = - 3$

Step 9.11

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

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

Step 9.12

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

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Step 9.12.1

Rewrite $- 3$ as ${(- 1)}^{3} \cdot 3$ .

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Step 9.12.1.1

Rewrite $- 3$ as $- 1 (3)$ .

$x = \sqrt[3]{- 1 (3)}$

Step 9.12.1.2

Rewrite $- 1$ as ${(- 1)}^{3}$ .

$x = \sqrt[3]{{(- 1)}^{3} \cdot 3}$

Step 9.12.2

Pull terms out from under the radical.

$x = - 1 \sqrt[3]{3}$

Step 9.12.3

Rewrite $- 1 \sqrt[3]{3}$ as $- \sqrt[3]{3}$ .

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

Step 9.13

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

$x = 1, - \frac{1 - i \sqrt{3}}{2}, - \frac{1 + i \sqrt{3}}{2}, - \sqrt[3]{3}$

Step 10