Trigonometry Examples

$\frac{x^{3} - 3 x^{2} - x - 10}{x^{2} + 3 x - 1}$

Step 1

Set the denominator in

$\frac{x^{3} - 3 x^{2} - x - 10}{x^{2} + 3 x - 1}$ equal to

$0$ to find where the expression is undefined.

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

Step 2

Solve for

$x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

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

Step 2.2

Substitute the values

$a = 1$ ,

$b = 3$ , and

$c = - 1$ into the quadratic formula and solve for

$x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot - 1)}}{2 \cdot 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Raise

$3$ to the power of

$2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 2.3.1.2

Multiply

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

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Step 2.3.1.2.1

Multiply

$- 4$ by

$1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.3.1.2.2

Multiply

$- 4$ by

$- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4}}{2 \cdot 1}$

Step 2.3.1.3

Add

$9$ and

$4$ .

$x = \frac{- 3 \pm \sqrt{13}}{2 \cdot 1}$

Step 2.3.2

Multiply

$2$ by

$1$ .

$x = \frac{- 3 \pm \sqrt{13}}{2}$

Step 2.4

Simplify the expression to solve for the

$+$ portion of the

$\pm$ .

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise

$3$ to the power of

$2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 2.4.1.2

Multiply

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

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Step 2.4.1.2.1

Multiply

$- 4$ by

$1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.4.1.2.2

Multiply

$- 4$ by

$- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4}}{2 \cdot 1}$

Step 2.4.1.3

Add

$9$ and

$4$ .

$x = \frac{- 3 \pm \sqrt{13}}{2 \cdot 1}$

Step 2.4.2

Multiply

$2$ by

$1$ .

$x = \frac{- 3 \pm \sqrt{13}}{2}$

Step 2.4.3

Change the

$\pm$ to

$+$ .

$x = \frac{- 3 + \sqrt{13}}{2}$

Step 2.4.4

Rewrite

$- 3$ as

$- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{13}}{2}$

Step 2.4.5

Factor

$- 1$ out of

$\sqrt{13}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{13})}{2}$

Step 2.4.6

Factor

$- 1$ out of

$- 1 (3) - 1 (- \sqrt{13})$ .

$x = \frac{- 1 (3 - \sqrt{13})}{2}$

Step 2.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{13}}{2}$

Step 2.5

Simplify the expression to solve for the

$-$ portion of the

$\pm$ .

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise

$3$ to the power of

$2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot - 1}}{2 \cdot 1}$

Step 2.5.1.2

Multiply

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

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Step 2.5.1.2.1

Multiply

$- 4$ by

$1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot - 1}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply

$- 4$ by

$- 1$ .

$x = \frac{- 3 \pm \sqrt{9 + 4}}{2 \cdot 1}$

Step 2.5.1.3

Add

$9$ and

$4$ .

$x = \frac{- 3 \pm \sqrt{13}}{2 \cdot 1}$

Step 2.5.2

Multiply

$2$ by

$1$ .

$x = \frac{- 3 \pm \sqrt{13}}{2}$

Step 2.5.3

Change the

$\pm$ to

$-$ .

$x = \frac{- 3 - \sqrt{13}}{2}$

Step 2.5.4

Rewrite

$- 3$ as

$- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{13}}{2}$

Step 2.5.5

Factor

$- 1$ out of

$- \sqrt{13}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{13})}{2}$

Step 2.5.6

Factor

$- 1$ out of

$- 1 (3) - (\sqrt{13})$ .

$x = \frac{- 1 (3 + \sqrt{13})}{2}$

Step 2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{13}}{2}$

Step 2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{13}}{2}, - \frac{3 + \sqrt{13}}{2}$

Step 3

The domain is all values of

$x$ that make the expression defined.

Interval Notation:

$(- \infty, - \frac{3 + \sqrt{13}}{2}) \cup (- \frac{3 + \sqrt{13}}{2}, - \frac{3 - \sqrt{13}}{2}) \cup (- \frac{3 - \sqrt{13}}{2}, \infty)$

Set-Builder Notation:

${x | x \neq - \frac{3 - \sqrt{13}}{2}, - \frac{3 + \sqrt{13}}{2}}$

Step 4