Trigonometry Examples

$(\frac{x^{2} - 9 x + 14}{x^{2} - 4}) (\frac{x^{2} + 3 x + 2}{x^{2} - 6 x - 7})$

Step 1

Find the domain to determine if the expression is continuous.

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Step 1.1

Set the denominator in $\frac{x^{2} - 9 x + 14}{x^{2} - 4}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 4 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Add $4$ to both sides of the equation.

$x^{2} = 4$

Step 1.2.2

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

$x = \pm \sqrt{4}$

Step 1.2.3

Simplify $\pm \sqrt{4}$ .

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Step 1.2.3.1

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

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

Step 1.2.3.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

Step 1.2.4

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

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Step 1.2.4.1

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

$x = 2$

Step 1.2.4.2

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

$x = - 2$

Step 1.2.4.3

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

$x = 2, - 2$

Step 1.3

Set the denominator in $\frac{x^{2} + 3 x + 2}{x^{2} - 6 x - 7}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 6 x - 7 = 0$

Step 1.4

Solve for $x$ .

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Step 1.4.1

Factor $x^{2} - 6 x - 7$ using the AC method.

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Step 1.4.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 7$ and whose sum is $- 6$ .

$- 7, 1$

Step 1.4.1.2

Write the factored form using these integers.

$(x - 7) (x + 1) = 0$

Step 1.4.2

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

$x - 7 = 0$

$x + 1 = 0$

Step 1.4.3

Set $x - 7$ equal to $0$ and solve for $x$ .

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Step 1.4.3.1

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Step 1.4.3.2

Add $7$ to both sides of the equation.

$x = 7$

Step 1.4.4

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

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Step 1.4.4.1

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

$x + 1 = 0$

Step 1.4.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.4.5

The final solution is all the values that make $(x - 7) (x + 1) = 0$ true.

$x = 7, - 1$

Step 1.5

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

Interval Notation:

$(- \infty, - 2) \cup (- 2, - 1) \cup (- 1, 2) \cup (2, 7) \cup (7, \infty)$

Set-Builder Notation:

${x | x \neq - 1, 2, - 2, 7}$

Interval Notation:

$(- \infty, - 2) \cup (- 2, - 1) \cup (- 1, 2) \cup (2, 7) \cup (7, \infty)$

Set-Builder Notation:

${x | x \neq - 1, 2, - 2, 7}$

Step 2

Since the domain is not all real numbers, $(\frac{x^{2} - 9 x + 14}{x^{2} - 4}) (\frac{x^{2} + 3 x + 2}{x^{2} - 6 x - 7})$ is not continuous over all real numbers.

Not continuous

Step 3