Linear Algebra Examples

$\frac{n - 10}{4 n^{2}} \cdot \frac{n^{2} - 36}{n^{2} - 4 n - 60}$

Step 1

Set the denominator in $\frac{n - 10}{4 n^{2}}$ equal to $0$ to find where the expression is undefined.

$4 n^{2} = 0$

Step 2

Solve for $n$ .

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Step 2.1

Divide each term in $4 n^{2} = 0$ by $4$ and simplify.

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Step 2.1.1

Divide each term in $4 n^{2} = 0$ by $4$ .

$\frac{4 n^{2}}{4} = \frac{0}{4}$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Cancel the common factor of $4$ .

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Step 2.1.2.1.1

Cancel the common factor.

$\frac{4 n^{2}}{4} = \frac{0}{4}$

Step 2.1.2.1.2

Divide $n^{2}$ by $1$ .

$n^{2} = \frac{0}{4}$

Step 2.1.3

Simplify the right side.

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Step 2.1.3.1

Divide $0$ by $4$ .

$n^{2} = 0$

Step 2.2

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

$n = \pm \sqrt{0}$

Step 2.3

Simplify $\pm \sqrt{0}$ .

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Step 2.3.1

Rewrite $0$ as $0^{2}$ .

$n = \pm \sqrt{0^{2}}$

Step 2.3.2

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

$n = \pm 0$

Step 2.3.3

Plus or minus $0$ is $0$ .

$n = 0$

Step 3

Set the denominator in $\frac{n^{2} - 36}{n^{2} - 4 n - 60}$ equal to $0$ to find where the expression is undefined.

$n^{2} - 4 n - 60 = 0$

Step 4

Solve for $n$ .

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Step 4.1

Factor $n^{2} - 4 n - 60$ using the AC method.

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Step 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 $- 60$ and whose sum is $- 4$ .

$- 10, 6$

Step 4.1.2

Write the factored form using these integers.

$(n - 10) (n + 6) = 0$

Step 4.2

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

$n - 10 = 0$

$n + 6 = 0$

Step 4.3

Set $n - 10$ equal to $0$ and solve for $n$ .

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Step 4.3.1

Set $n - 10$ equal to $0$ .

$n - 10 = 0$

Step 4.3.2

Add $10$ to both sides of the equation.

$n = 10$

Step 4.4

Set $n + 6$ equal to $0$ and solve for $n$ .

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Step 4.4.1

Set $n + 6$ equal to $0$ .

$n + 6 = 0$

Step 4.4.2

Subtract $6$ from both sides of the equation.

$n = - 6$

Step 4.5

The final solution is all the values that make $(n - 10) (n + 6) = 0$ true.

$n = 10, - 6$

Step 5

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

Interval Notation:

$(- \infty, - 6) \cup (- 6, 0) \cup (0, 10) \cup (10, \infty)$

Set-Builder Notation:

${n | n \neq 0, - 6, 10}$

Step 6