Trigonometry Examples

$\frac{- 3 x^{2} - 48 x - 4}{(x + 9) (x^{2} + 8)}$

Step 1

Set the denominator in $\frac{- 3 x^{2} - 48 x - 4}{(x + 9) (x^{2} + 8)}$ equal to $0$ to find where the expression is undefined.

$(x + 9) (x^{2} + 8) = 0$

Step 2

Solve for $x$ .

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Step 2.1

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

$x + 9 = 0$

$x^{2} + 8 = 0$

Step 2.2

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 2.2.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 2.2.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 2.3

Set $x^{2} + 8$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x^{2} + 8$ equal to $0$ .

$x^{2} + 8 = 0$

Step 2.3.2

Solve $x^{2} + 8 = 0$ for $x$ .

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Step 2.3.2.1

Subtract $8$ from both sides of the equation.

$x^{2} = - 8$

Step 2.3.2.2

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

$x = \pm \sqrt{- 8}$

Step 2.3.2.3

Simplify $\pm \sqrt{- 8}$ .

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Step 2.3.2.3.1

Rewrite $- 8$ as $- 1 (8)$ .

$x = \pm \sqrt{- 1 (8)}$

Step 2.3.2.3.2

Rewrite $\sqrt{- 1 (8)}$ as $\sqrt{- 1} \cdot \sqrt{8}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{8}$

Step 2.3.2.3.3

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

$x = \pm i \cdot \sqrt{8}$

Step 2.3.2.3.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 2.3.2.3.4.1

Factor $4$ out of $8$ .

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

Step 2.3.2.3.4.2

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

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

Step 2.3.2.3.5

Pull terms out from under the radical.

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

Step 2.3.2.3.6

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

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

Step 2.3.2.4

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

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Step 2.3.2.4.1

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

$x = 2 i \sqrt{2}$

Step 2.3.2.4.2

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

$x = - 2 i \sqrt{2}$

Step 2.3.2.4.3

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

$x = 2 i \sqrt{2}, - 2 i \sqrt{2}$

Step 2.4

The final solution is all the values that make $(x + 9) (x^{2} + 8) = 0$ true.

$x = - 9, 2 i \sqrt{2}, - 2 i \sqrt{2}$

Step 3

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

Interval Notation:

$(- \infty, - 9) \cup (- 9, \infty)$

Set-Builder Notation:

${x | x \neq - 9}$

Step 4