Algebra Examples

$\frac{n^{2} + 4 n - 12}{2 n^{4} - 72 n^{2}} \cdot \frac{5 n^{3} - 30 n^{2}}{11 n - 22}$

Step 1

Set the denominator in $\frac{n^{2} + 4 n - 12}{2 n^{4} - 72 n^{2}}$ equal to $0$ to find where the expression is undefined.

$2 n^{4} - 72 n^{2} = 0$

Step 2

Solve for $n$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Rewrite $n^{4}$ as ${(n^{2})}^{2}$ .

$2 {(n^{2})}^{2} - 72 n^{2} = 0$

Step 2.1.2

Let $u = n^{2}$ . Substitute $u$ for all occurrences of $n^{2}$ .

$2 u^{2} - 72 u = 0$

Step 2.1.3

Factor $2 u$ out of $2 u^{2} - 72 u$ .

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Step 2.1.3.1

Factor $2 u$ out of $2 u^{2}$ .

$2 u (u) - 72 u = 0$

Step 2.1.3.2

Factor $2 u$ out of $- 72 u$ .

$2 u (u) + 2 u (- 36) = 0$

Step 2.1.3.3

Factor $2 u$ out of $2 u (u) + 2 u (- 36)$ .

$2 u (u - 36) = 0$

Step 2.1.4

Replace all occurrences of $u$ with $n^{2}$ .

$2 n^{2} (n^{2} - 36) = 0$

Step 2.2

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

$n^{2} = 0$

$n^{2} - 36 = 0$

Step 2.3

Set $n^{2}$ equal to $0$ and solve for $n$ .

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Step 2.3.1

Set $n^{2}$ equal to $0$ .

$n^{2} = 0$

Step 2.3.2

Solve $n^{2} = 0$ for $n$ .

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Step 2.3.2.1

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

$n = \pm \sqrt{0}$

Step 2.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

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

$n = \pm 0$

Step 2.3.2.2.3

Plus or minus $0$ is $0$ .

$n = 0$

Step 2.4

Set $n^{2} - 36$ equal to $0$ and solve for $n$ .

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Step 2.4.1

Set $n^{2} - 36$ equal to $0$ .

$n^{2} - 36 = 0$

Step 2.4.2

Solve $n^{2} - 36 = 0$ for $n$ .

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Step 2.4.2.1

Add $36$ to both sides of the equation.

$n^{2} = 36$

Step 2.4.2.2

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

$n = \pm \sqrt{36}$

Step 2.4.2.3

Simplify $\pm \sqrt{36}$ .

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Step 2.4.2.3.1

Rewrite $36$ as $6^{2}$ .

$n = \pm \sqrt{6^{2}}$

Step 2.4.2.3.2

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

$n = \pm 6$

Step 2.4.2.4

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

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Step 2.4.2.4.1

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

$n = 6$

Step 2.4.2.4.2

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

$n = - 6$

Step 2.4.2.4.3

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

$n = 6, - 6$

Step 2.5

The final solution is all the values that make $2 n^{2} (n^{2} - 36) = 0$ true.

$n = 0, 6, - 6$

Step 3

Set the denominator in $\frac{5 n^{3} - 30 n^{2}}{11 n - 22}$ equal to $0$ to find where the expression is undefined.

$11 n - 22 = 0$

Step 4

Solve for $n$ .

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Step 4.1

Add $22$ to both sides of the equation.

$11 n = 22$

Step 4.2

Divide each term in $11 n = 22$ by $11$ and simplify.

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Step 4.2.1

Divide each term in $11 n = 22$ by $11$ .

$\frac{11 n}{11} = \frac{22}{11}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $11$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{11 n}{11} = \frac{22}{11}$

Step 4.2.2.1.2

Divide $n$ by $1$ .

$n = \frac{22}{11}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $22$ by $11$ .

$n = 2$

Step 5

The equation is undefined where the denominator equals $0$ , the argument of a square root is less than $0$ , or the argument of a logarithm is less than or equal to $0$ .

$n = - 6, n = 0, n = 2, n = 6$

Step 6