Finite Math Examples

$f (x) = 3 x + 30$ , $g (x) = x^{2} + 20$

Step 1

Find the quotient of the functions.

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Step 1.1

Replace the function designators with the actual functions in $\frac{f (x)}{g (x)}$ .

$\frac{3 x + 30}{x^{2} + 20}$

Step 1.2

Factor $3$ out of $3 x + 30$ .

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Step 1.2.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) + 30}{x^{2} + 20}$

Step 1.2.2

Factor $3$ out of $30$ .

$\frac{3 x + 3 \cdot 10}{x^{2} + 20}$

Step 1.2.3

Factor $3$ out of $3 x + 3 \cdot 10$ .

$\frac{3 (x + 10)}{x^{2} + 20}$

Step 2

Set the denominator in $\frac{3 (x + 10)}{x^{2} + 20}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 20 = 0$

Step 3

Solve for $x$ .

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Step 3.1

Subtract $20$ from both sides of the equation.

$x^{2} = - 20$

Step 3.2

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

$x = \pm \sqrt{- 20}$

Step 3.3

Simplify $\pm \sqrt{- 20}$ .

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Step 3.3.1

Rewrite $- 20$ as $- 1 (20)$ .

$x = \pm \sqrt{- 1 (20)}$

Step 3.3.2

Rewrite $\sqrt{- 1 (20)}$ as $\sqrt{- 1} \cdot \sqrt{20}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{20}$

Step 3.3.3

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

$x = \pm i \cdot \sqrt{20}$

Step 3.3.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 3.3.4.1

Factor $4$ out of $20$ .

$x = \pm i \cdot \sqrt{4 (5)}$

Step 3.3.4.2

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

$x = \pm i \cdot \sqrt{2^{2} \cdot 5}$

Step 3.3.5

Pull terms out from under the radical.

$x = \pm i \cdot (2 \sqrt{5})$

Step 3.3.6

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

$x = \pm 2 i \sqrt{5}$

Step 3.4

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

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Step 3.4.1

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

$x = 2 i \sqrt{5}$

Step 3.4.2

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

$x = - 2 i \sqrt{5}$

Step 3.4.3

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

$x = 2 i \sqrt{5}, - 2 i \sqrt{5}$

Step 4

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 5