Finite Math Examples

${(x + 2)}^{3} = x^{2} + 8$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $x^{2}$ from both sides of the equation.

${(x + 2)}^{3} - x^{2} = 8$

Step 1.2

Subtract $8$ from both sides of the equation.

${(x + 2)}^{3} - x^{2} - 8 = 0$

Step 2

Simplify ${(x + 2)}^{3} - x^{2} - 8$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Use the Binomial Theorem.

$x^{3} + 3 x^{2} \cdot 2 + 3 x \cdot 2^{2} + 2^{3} - x^{2} - 8 = 0$

Step 2.1.2

Simplify each term.

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Step 2.1.2.1

Multiply $2$ by $3$ .

$x^{3} + 6 x^{2} + 3 x \cdot 2^{2} + 2^{3} - x^{2} - 8 = 0$

Step 2.1.2.2

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

$x^{3} + 6 x^{2} + 3 x \cdot 4 + 2^{3} - x^{2} - 8 = 0$

Step 2.1.2.3

Multiply $4$ by $3$ .

$x^{3} + 6 x^{2} + 12 x + 2^{3} - x^{2} - 8 = 0$

Step 2.1.2.4

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

$x^{3} + 6 x^{2} + 12 x + 8 - x^{2} - 8 = 0$

Step 2.2

Combine the opposite terms in $x^{3} + 6 x^{2} + 12 x + 8 - x^{2} - 8$ .

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Step 2.2.1

Subtract $8$ from $8$ .

$x^{3} + 6 x^{2} + 12 x - x^{2} + 0 = 0$

Step 2.2.2

Add $x^{3} + 6 x^{2} + 12 x - x^{2}$ and $0$ .

$x^{3} + 6 x^{2} + 12 x - x^{2} = 0$

Step 2.3

Subtract $x^{2}$ from $6 x^{2}$ .

$x^{3} + 5 x^{2} + 12 x = 0$

Step 3

Factor $x$ out of $x^{3} + 5 x^{2} + 12 x$ .

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Step 3.1

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

$x \cdot x^{2} + 5 x^{2} + 12 x = 0$

Step 3.2

Factor $x$ out of $5 x^{2}$ .

$x \cdot x^{2} + x (5 x) + 12 x = 0$

Step 3.3

Factor $x$ out of $12 x$ .

$x \cdot x^{2} + x (5 x) + x \cdot 12 = 0$

Step 3.4

Factor $x$ out of $x \cdot x^{2} + x (5 x)$ .

$x (x^{2} + 5 x) + x \cdot 12 = 0$

Step 3.5

Factor $x$ out of $x (x^{2} + 5 x) + x \cdot 12$ .

$x (x^{2} + 5 x + 12) = 0$

Step 4

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

$x = 0$

$x^{2} + 5 x + 12 = 0$

Step 5

Set $x$ equal to $0$ .

$x = 0$

Step 6

Set $x^{2} + 5 x + 12$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{2} + 5 x + 12$ equal to $0$ .

$x^{2} + 5 x + 12 = 0$

Step 6.2

Solve $x^{2} + 5 x + 12 = 0$ for $x$ .

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Step 6.2.1

Use the quadratic formula to find the solutions.

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

Step 6.2.2

Substitute the values $a = 1$ , $b = 5$ , and $c = 12$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (1 \cdot 12)}}{2 \cdot 1}$

Step 6.2.3

Simplify.

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Step 6.2.3.1

Simplify the numerator.

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Step 6.2.3.1.1

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Step 6.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 12$ .

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Step 6.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 12}}{2 \cdot 1}$

Step 6.2.3.1.2.2

Multiply $- 4$ by $12$ .

$x = \frac{- 5 \pm \sqrt{25 - 48}}{2 \cdot 1}$

Step 6.2.3.1.3

Subtract $48$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 23}}{2 \cdot 1}$

Step 6.2.3.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 1}$

Step 6.2.3.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 1}$

Step 6.2.3.1.6

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

$x = \frac{- 5 \pm i \sqrt{23}}{2 \cdot 1}$

Step 6.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm i \sqrt{23}}{2}$

Step 6.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 6.2.4.1

Simplify the numerator.

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Step 6.2.4.1.1

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Step 6.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 12$ .

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Step 6.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 12}}{2 \cdot 1}$

Step 6.2.4.1.2.2

Multiply $- 4$ by $12$ .

$x = \frac{- 5 \pm \sqrt{25 - 48}}{2 \cdot 1}$

Step 6.2.4.1.3

Subtract $48$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 23}}{2 \cdot 1}$

Step 6.2.4.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 1}$

Step 6.2.4.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 1}$

Step 6.2.4.1.6

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

$x = \frac{- 5 \pm i \sqrt{23}}{2 \cdot 1}$

Step 6.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm i \sqrt{23}}{2}$

Step 6.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 5 + i \sqrt{23}}{2}$

Step 6.2.4.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 + i \sqrt{23}}{2}$

Step 6.2.4.5

Factor $- 1$ out of $i \sqrt{23}$ .

$x = \frac{- 1 \cdot 5 - (- i \sqrt{23})}{2}$

Step 6.2.4.6

Factor $- 1$ out of $- 1 (5) - (- i \sqrt{23})$ .

$x = \frac{- 1 (5 - i \sqrt{23})}{2}$

Step 6.2.4.7

Move the negative in front of the fraction.

$x = - \frac{5 - i \sqrt{23}}{2}$

Step 6.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 6.2.5.1

Simplify the numerator.

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Step 6.2.5.1.1

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 1 \cdot 12}}{2 \cdot 1}$

Step 6.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 12$ .

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Step 6.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 12}}{2 \cdot 1}$

Step 6.2.5.1.2.2

Multiply $- 4$ by $12$ .

$x = \frac{- 5 \pm \sqrt{25 - 48}}{2 \cdot 1}$

Step 6.2.5.1.3

Subtract $48$ from $25$ .

$x = \frac{- 5 \pm \sqrt{- 23}}{2 \cdot 1}$

Step 6.2.5.1.4

Rewrite $- 23$ as $- 1 (23)$ .

$x = \frac{- 5 \pm \sqrt{- 1 \cdot 23}}{2 \cdot 1}$

Step 6.2.5.1.5

Rewrite $\sqrt{- 1 (23)}$ as $\sqrt{- 1} \cdot \sqrt{23}$ .

$x = \frac{- 5 \pm \sqrt{- 1} \cdot \sqrt{23}}{2 \cdot 1}$

Step 6.2.5.1.6

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

$x = \frac{- 5 \pm i \sqrt{23}}{2 \cdot 1}$

Step 6.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 5 \pm i \sqrt{23}}{2}$

Step 6.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 5 - i \sqrt{23}}{2}$

Step 6.2.5.4

Rewrite $- 5$ as $- 1 (5)$ .

$x = \frac{- 1 \cdot 5 - i \sqrt{23}}{2}$

Step 6.2.5.5

Factor $- 1$ out of $- i \sqrt{23}$ .

$x = \frac{- 1 \cdot 5 - (i \sqrt{23})}{2}$

Step 6.2.5.6

Factor $- 1$ out of $- 1 (5) - (i \sqrt{23})$ .

$x = \frac{- 1 (5 + i \sqrt{23})}{2}$

Step 6.2.5.7

Move the negative in front of the fraction.

$x = - \frac{5 + i \sqrt{23}}{2}$

Step 6.2.6

The final answer is the combination of both solutions.

$x = - \frac{5 - i \sqrt{23}}{2}, - \frac{5 + i \sqrt{23}}{2}$

Step 7

The final solution is all the values that make $x (x^{2} + 5 x + 12) = 0$ true.

$x = 0, - \frac{5 - i \sqrt{23}}{2}, - \frac{5 + i \sqrt{23}}{2}$