Finite Math Examples

$r (x) = 400 x$ , $c (x) = x^{2} - 100 x + 3000$

Step 1

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

$\frac{400 x}{x^{2} - 100 x + 3000}$

Step 2

Set the denominator in $\frac{400 x}{x^{2} - 100 x + 3000}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 100 x + 3000 = 0$

Step 3

Solve for $x$ .

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Step 3.1

Use the quadratic formula to find the solutions.

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

Step 3.2

Substitute the values $a = 1$ , $b = - 100$ , and $c = 3000$ into the quadratic formula and solve for $x$ .

$\frac{100 \pm \sqrt{{(- 100)}^{2} - 4 \cdot (1 \cdot 3000)}}{2 \cdot 1}$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Raise $- 100$ to the power of $2$ .

$x = \frac{100 \pm \sqrt{10000 - 4 \cdot 1 \cdot 3000}}{2 \cdot 1}$

Step 3.3.1.2

Multiply $- 4 \cdot 1 \cdot 3000$ .

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Step 3.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{100 \pm \sqrt{10000 - 4 \cdot 3000}}{2 \cdot 1}$

Step 3.3.1.2.2

Multiply $- 4$ by $3000$ .

$x = \frac{100 \pm \sqrt{10000 - 12000}}{2 \cdot 1}$

Step 3.3.1.3

Subtract $12000$ from $10000$ .

$x = \frac{100 \pm \sqrt{- 2000}}{2 \cdot 1}$

Step 3.3.1.4

Rewrite $- 2000$ as $- 1 (2000)$ .

$x = \frac{100 \pm \sqrt{- 1 \cdot 2000}}{2 \cdot 1}$

Step 3.3.1.5

Rewrite $\sqrt{- 1 (2000)}$ as $\sqrt{- 1} \cdot \sqrt{2000}$ .

$x = \frac{100 \pm \sqrt{- 1} \cdot \sqrt{2000}}{2 \cdot 1}$

Step 3.3.1.6

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

$x = \frac{100 \pm i \cdot \sqrt{2000}}{2 \cdot 1}$

Step 3.3.1.7

Rewrite $2000$ as $20^{2} \cdot 5$ .

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Step 3.3.1.7.1

Factor $400$ out of $2000$ .

$x = \frac{100 \pm i \cdot \sqrt{400 (5)}}{2 \cdot 1}$

Step 3.3.1.7.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{100 \pm i \cdot \sqrt{20^{2} \cdot 5}}{2 \cdot 1}$

Step 3.3.1.8

Pull terms out from under the radical.

$x = \frac{100 \pm i \cdot (20 \sqrt{5})}{2 \cdot 1}$

Step 3.3.1.9

Move $20$ to the left of $i$ .

$x = \frac{100 \pm 20 i \sqrt{5}}{2 \cdot 1}$

Step 3.3.2

Multiply $2$ by $1$ .

$x = \frac{100 \pm 20 i \sqrt{5}}{2}$

Step 3.3.3

Simplify $\frac{100 \pm 20 i \sqrt{5}}{2}$ .

$x = 50 \pm 10 i \sqrt{5}$

Step 3.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 3.4.1

Simplify the numerator.

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Step 3.4.1.1

Raise $- 100$ to the power of $2$ .

$x = \frac{100 \pm \sqrt{10000 - 4 \cdot 1 \cdot 3000}}{2 \cdot 1}$

Step 3.4.1.2

Multiply $- 4 \cdot 1 \cdot 3000$ .

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Step 3.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{100 \pm \sqrt{10000 - 4 \cdot 3000}}{2 \cdot 1}$

Step 3.4.1.2.2

Multiply $- 4$ by $3000$ .

$x = \frac{100 \pm \sqrt{10000 - 12000}}{2 \cdot 1}$

Step 3.4.1.3

Subtract $12000$ from $10000$ .

$x = \frac{100 \pm \sqrt{- 2000}}{2 \cdot 1}$

Step 3.4.1.4

Rewrite $- 2000$ as $- 1 (2000)$ .

$x = \frac{100 \pm \sqrt{- 1 \cdot 2000}}{2 \cdot 1}$

Step 3.4.1.5

Rewrite $\sqrt{- 1 (2000)}$ as $\sqrt{- 1} \cdot \sqrt{2000}$ .

$x = \frac{100 \pm \sqrt{- 1} \cdot \sqrt{2000}}{2 \cdot 1}$

Step 3.4.1.6

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

$x = \frac{100 \pm i \cdot \sqrt{2000}}{2 \cdot 1}$

Step 3.4.1.7

Rewrite $2000$ as $20^{2} \cdot 5$ .

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Step 3.4.1.7.1

Factor $400$ out of $2000$ .

$x = \frac{100 \pm i \cdot \sqrt{400 (5)}}{2 \cdot 1}$

Step 3.4.1.7.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{100 \pm i \cdot \sqrt{20^{2} \cdot 5}}{2 \cdot 1}$

Step 3.4.1.8

Pull terms out from under the radical.

$x = \frac{100 \pm i \cdot (20 \sqrt{5})}{2 \cdot 1}$

Step 3.4.1.9

Move $20$ to the left of $i$ .

$x = \frac{100 \pm 20 i \sqrt{5}}{2 \cdot 1}$

Step 3.4.2

Multiply $2$ by $1$ .

$x = \frac{100 \pm 20 i \sqrt{5}}{2}$

Step 3.4.3

Simplify $\frac{100 \pm 20 i \sqrt{5}}{2}$ .

$x = 50 \pm 10 i \sqrt{5}$

Step 3.4.4

Change the $\pm$ to $+$ .

$x = 50 + 10 i \sqrt{5}$

Step 3.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $- 100$ to the power of $2$ .

$x = \frac{100 \pm \sqrt{10000 - 4 \cdot 1 \cdot 3000}}{2 \cdot 1}$

Step 3.5.1.2

Multiply $- 4 \cdot 1 \cdot 3000$ .

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Step 3.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{100 \pm \sqrt{10000 - 4 \cdot 3000}}{2 \cdot 1}$

Step 3.5.1.2.2

Multiply $- 4$ by $3000$ .

$x = \frac{100 \pm \sqrt{10000 - 12000}}{2 \cdot 1}$

Step 3.5.1.3

Subtract $12000$ from $10000$ .

$x = \frac{100 \pm \sqrt{- 2000}}{2 \cdot 1}$

Step 3.5.1.4

Rewrite $- 2000$ as $- 1 (2000)$ .

$x = \frac{100 \pm \sqrt{- 1 \cdot 2000}}{2 \cdot 1}$

Step 3.5.1.5

Rewrite $\sqrt{- 1 (2000)}$ as $\sqrt{- 1} \cdot \sqrt{2000}$ .

$x = \frac{100 \pm \sqrt{- 1} \cdot \sqrt{2000}}{2 \cdot 1}$

Step 3.5.1.6

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

$x = \frac{100 \pm i \cdot \sqrt{2000}}{2 \cdot 1}$

Step 3.5.1.7

Rewrite $2000$ as $20^{2} \cdot 5$ .

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Step 3.5.1.7.1

Factor $400$ out of $2000$ .

$x = \frac{100 \pm i \cdot \sqrt{400 (5)}}{2 \cdot 1}$

Step 3.5.1.7.2

Rewrite $400$ as $20^{2}$ .

$x = \frac{100 \pm i \cdot \sqrt{20^{2} \cdot 5}}{2 \cdot 1}$

Step 3.5.1.8

Pull terms out from under the radical.

$x = \frac{100 \pm i \cdot (20 \sqrt{5})}{2 \cdot 1}$

Step 3.5.1.9

Move $20$ to the left of $i$ .

$x = \frac{100 \pm 20 i \sqrt{5}}{2 \cdot 1}$

Step 3.5.2

Multiply $2$ by $1$ .

$x = \frac{100 \pm 20 i \sqrt{5}}{2}$

Step 3.5.3

Simplify $\frac{100 \pm 20 i \sqrt{5}}{2}$ .

$x = 50 \pm 10 i \sqrt{5}$

Step 3.5.4

Change the $\pm$ to $-$ .

$x = 50 - 10 i \sqrt{5}$

Step 3.6

The final answer is the combination of both solutions.

$x = 50 + 10 i \sqrt{5}, 50 - 10 i \sqrt{5}$

Step 4

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 5