Algebra Examples

$f (x) = - \sqrt{x^{3} + 8}$

Step 1

Set the radicand in $\sqrt{x^{3} + 8}$ greater than or equal to $0$ to find where the expression is defined.

$x^{3} + 8 \geq 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $8$ from both sides of the inequality.

$x^{3} \geq - 8$

Step 2.2

Add $8$ to both sides of the inequality.

$x^{3} + 8 \geq 0$

Step 2.3

Convert the inequality to an equation.

$x^{3} + 8 = 0$

Step 2.4

Factor the left side of the equation.

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Step 2.4.1

Rewrite $8$ as $2^{3}$ .

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

Step 2.4.2

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

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

Step 2.4.3

Simplify.

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Step 2.4.3.1

Multiply $2$ by $- 1$ .

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

Step 2.4.3.2

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

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

Step 2.5

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

$x + 2 = 0$

$x^{2} - 2 x + 4 = 0$

Step 2.6

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 2.6.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 2.7

Set $x^{2} - 2 x + 4$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $x^{2} - 2 x + 4$ equal to $0$ .

$x^{2} - 2 x + 4 = 0$

Step 2.7.2

Solve $x^{2} - 2 x + 4 = 0$ for $x$ .

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Step 2.7.2.1

Use the quadratic formula to find the solutions.

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

Step 2.7.2.2

Substitute the values $a = 1$ , $b = - 2$ , and $c = 4$ into the quadratic formula and solve for $x$ .

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

Step 2.7.2.3

Simplify.

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Step 2.7.2.3.1

Simplify the numerator.

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Step 2.7.2.3.1.1

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

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

Step 2.7.2.3.1.2

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

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Step 2.7.2.3.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.7.2.3.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 2.7.2.3.1.3

Subtract $16$ from $4$ .

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

Step 2.7.2.3.1.4

Rewrite $- 12$ as $- 1 (12)$ .

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

Step 2.7.2.3.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

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

Step 2.7.2.3.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.7.2.3.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.7.2.3.1.7.1

Factor $4$ out of $12$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.7.2.3.1.7.2

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

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

Step 2.7.2.3.1.8

Pull terms out from under the radical.

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

Step 2.7.2.3.1.9

Move $2$ to the left of $i$ .

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

Step 2.7.2.3.2

Multiply $2$ by $1$ .

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

Step 2.7.2.3.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{2}$ .

$x = 1 \pm i \sqrt{3}$

Step 2.7.2.4

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

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Step 2.7.2.4.1

Simplify the numerator.

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Step 2.7.2.4.1.1

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

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

Step 2.7.2.4.1.2

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

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Step 2.7.2.4.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.7.2.4.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 2.7.2.4.1.3

Subtract $16$ from $4$ .

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

Step 2.7.2.4.1.4

Rewrite $- 12$ as $- 1 (12)$ .

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

Step 2.7.2.4.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

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

Step 2.7.2.4.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.7.2.4.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.7.2.4.1.7.1

Factor $4$ out of $12$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.7.2.4.1.7.2

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

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

Step 2.7.2.4.1.8

Pull terms out from under the radical.

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

Step 2.7.2.4.1.9

Move $2$ to the left of $i$ .

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

Step 2.7.2.4.2

Multiply $2$ by $1$ .

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

Step 2.7.2.4.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{2}$ .

$x = 1 \pm i \sqrt{3}$

Step 2.7.2.4.4

Change the $\pm$ to $+$ .

$x = 1 + i \sqrt{3}$

Step 2.7.2.5

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

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Step 2.7.2.5.1

Simplify the numerator.

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Step 2.7.2.5.1.1

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

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

Step 2.7.2.5.1.2

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

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Step 2.7.2.5.1.2.1

Multiply $- 4$ by $1$ .

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

Step 2.7.2.5.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 2.7.2.5.1.3

Subtract $16$ from $4$ .

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

Step 2.7.2.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

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

Step 2.7.2.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

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

Step 2.7.2.5.1.6

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

$x = \frac{2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 2.7.2.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 2.7.2.5.1.7.1

Factor $4$ out of $12$ .

$x = \frac{2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 2.7.2.5.1.7.2

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

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

Step 2.7.2.5.1.8

Pull terms out from under the radical.

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

Step 2.7.2.5.1.9

Move $2$ to the left of $i$ .

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

Step 2.7.2.5.2

Multiply $2$ by $1$ .

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

Step 2.7.2.5.3

Simplify $\frac{2 \pm 2 i \sqrt{3}}{2}$ .

$x = 1 \pm i \sqrt{3}$

Step 2.7.2.5.4

Change the $\pm$ to $-$ .

$x = 1 - i \sqrt{3}$

Step 2.7.2.6

The final answer is the combination of both solutions.

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

Step 2.8

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

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

Step 2.9

The solution consists of all of the true intervals.

$x \geq - 2$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$[- 2, \infty)$

Set-Builder Notation:

${x | x \geq - 2}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$(- \infty, 0]$

Set-Builder Notation:

${y | y \leq 0}$

Step 5

Determine the domain and range.

Domain: $[- 2, \infty), {x | x \geq - 2}$

Range: $(- \infty, 0], {y | y \leq 0}$

Step 6