Finite Math Examples

$f (x) = \frac{5 x^{2} - 9 x - 9}{8 x^{2} - 2 x + 8}$

Step 1

Set the denominator in $\frac{5 x^{2} - 9 x - 9}{8 x^{2} - 2 x + 8}$ equal to $0$ to find where the expression is undefined.

$8 x^{2} - 2 x + 8 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor $2$ out of $8 x^{2} - 2 x + 8$ .

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Step 2.1.1

Factor $2$ out of $8 x^{2}$ .

$2 (4 x^{2}) - 2 x + 8 = 0$

Step 2.1.2

Factor $2$ out of $- 2 x$ .

$2 (4 x^{2}) + 2 (- x) + 8 = 0$

Step 2.1.3

Factor $2$ out of $8$ .

$2 (4 x^{2}) + 2 (- x) + 2 (4) = 0$

Step 2.1.4

Factor $2$ out of $2 (4 x^{2}) + 2 (- x)$ .

$2 (4 x^{2} - x) + 2 (4) = 0$

Step 2.1.5

Factor $2$ out of $2 (4 x^{2} - x) + 2 (4)$ .

$2 (4 x^{2} - x + 4) = 0$

Step 2.2

Divide each term in $2 (4 x^{2} - x + 4) = 0$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 (4 x^{2} - x + 4) = 0$ by $2$ .

$\frac{2 (4 x^{2} - x + 4)}{2} = \frac{0}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 (4 x^{2} - x + 4)}{2} = \frac{0}{2}$

Step 2.2.2.1.2

Divide $4 x^{2} - x + 4$ by $1$ .

$4 x^{2} - x + 4 = \frac{0}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Divide $0$ by $2$ .

$4 x^{2} - x + 4 = 0$

Step 2.3

Use the quadratic formula to find the solutions.

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

Step 2.4

Substitute the values $a = 4$ , $b = - 1$ , and $c = 4$ into the quadratic formula and solve for $x$ .

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

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 1$ to the power of $2$ .

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

Step 2.5.1.2

Multiply $- 4 \cdot 4 \cdot 4$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{1 \pm \sqrt{1 - 16 \cdot 4}}{2 \cdot 4}$

Step 2.5.1.2.2

Multiply $- 16$ by $4$ .

$x = \frac{1 \pm \sqrt{1 - 64}}{2 \cdot 4}$

Step 2.5.1.3

Subtract $64$ from $1$ .

$x = \frac{1 \pm \sqrt{- 63}}{2 \cdot 4}$

Step 2.5.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 4}$

Step 2.5.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 4}$

Step 2.5.1.6

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

$x = \frac{1 \pm i \cdot \sqrt{63}}{2 \cdot 4}$

Step 2.5.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 2.5.1.7.1

Factor $9$ out of $63$ .

$x = \frac{1 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 4}$

Step 2.5.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{1 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 4}$

Step 2.5.1.8

Pull terms out from under the radical.

$x = \frac{1 \pm i \cdot (3 \sqrt{7})}{2 \cdot 4}$

Step 2.5.1.9

Move $3$ to the left of $i$ .

$x = \frac{1 \pm 3 i \sqrt{7}}{2 \cdot 4}$

Step 2.5.2

Multiply $2$ by $4$ .

$x = \frac{1 \pm 3 i \sqrt{7}}{8}$

Step 2.6

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

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Step 2.6.1

Simplify the numerator.

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Step 2.6.1.1

Raise $- 1$ to the power of $2$ .

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

Step 2.6.1.2

Multiply $- 4 \cdot 4 \cdot 4$ .

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Step 2.6.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{1 \pm \sqrt{1 - 16 \cdot 4}}{2 \cdot 4}$

Step 2.6.1.2.2

Multiply $- 16$ by $4$ .

$x = \frac{1 \pm \sqrt{1 - 64}}{2 \cdot 4}$

Step 2.6.1.3

Subtract $64$ from $1$ .

$x = \frac{1 \pm \sqrt{- 63}}{2 \cdot 4}$

Step 2.6.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 4}$

Step 2.6.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 4}$

Step 2.6.1.6

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

$x = \frac{1 \pm i \cdot \sqrt{63}}{2 \cdot 4}$

Step 2.6.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 2.6.1.7.1

Factor $9$ out of $63$ .

$x = \frac{1 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 4}$

Step 2.6.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{1 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 4}$

Step 2.6.1.8

Pull terms out from under the radical.

$x = \frac{1 \pm i \cdot (3 \sqrt{7})}{2 \cdot 4}$

Step 2.6.1.9

Move $3$ to the left of $i$ .

$x = \frac{1 \pm 3 i \sqrt{7}}{2 \cdot 4}$

Step 2.6.2

Multiply $2$ by $4$ .

$x = \frac{1 \pm 3 i \sqrt{7}}{8}$

Step 2.6.3

Change the $\pm$ to $+$ .

$x = \frac{1 + 3 i \sqrt{7}}{8}$

Step 2.7

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

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Step 2.7.1

Simplify the numerator.

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Step 2.7.1.1

Raise $- 1$ to the power of $2$ .

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

Step 2.7.1.2

Multiply $- 4 \cdot 4 \cdot 4$ .

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Step 2.7.1.2.1

Multiply $- 4$ by $4$ .

$x = \frac{1 \pm \sqrt{1 - 16 \cdot 4}}{2 \cdot 4}$

Step 2.7.1.2.2

Multiply $- 16$ by $4$ .

$x = \frac{1 \pm \sqrt{1 - 64}}{2 \cdot 4}$

Step 2.7.1.3

Subtract $64$ from $1$ .

$x = \frac{1 \pm \sqrt{- 63}}{2 \cdot 4}$

Step 2.7.1.4

Rewrite $- 63$ as $- 1 (63)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 63}}{2 \cdot 4}$

Step 2.7.1.5

Rewrite $\sqrt{- 1 (63)}$ as $\sqrt{- 1} \cdot \sqrt{63}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{63}}{2 \cdot 4}$

Step 2.7.1.6

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

$x = \frac{1 \pm i \cdot \sqrt{63}}{2 \cdot 4}$

Step 2.7.1.7

Rewrite $63$ as $3^{2} \cdot 7$ .

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Step 2.7.1.7.1

Factor $9$ out of $63$ .

$x = \frac{1 \pm i \cdot \sqrt{9 (7)}}{2 \cdot 4}$

Step 2.7.1.7.2

Rewrite $9$ as $3^{2}$ .

$x = \frac{1 \pm i \cdot \sqrt{3^{2} \cdot 7}}{2 \cdot 4}$

Step 2.7.1.8

Pull terms out from under the radical.

$x = \frac{1 \pm i \cdot (3 \sqrt{7})}{2 \cdot 4}$

Step 2.7.1.9

Move $3$ to the left of $i$ .

$x = \frac{1 \pm 3 i \sqrt{7}}{2 \cdot 4}$

Step 2.7.2

Multiply $2$ by $4$ .

$x = \frac{1 \pm 3 i \sqrt{7}}{8}$

Step 2.7.3

Change the $\pm$ to $-$ .

$x = \frac{1 - 3 i \sqrt{7}}{8}$

Step 2.8

The final answer is the combination of both solutions.

$x = \frac{1 + 3 i \sqrt{7}}{8}, \frac{1 - 3 i \sqrt{7}}{8}$

Step 3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

The range is the set of all valid $y$ values. Use the graph to find the range.

Interval Notation:

$[- \frac{41 + 2 \sqrt{4531}}{126}, - \frac{41 - 2 \sqrt{4531}}{126}]$

Set-Builder Notation:

${y | - \frac{41 + 2 \sqrt{4531}}{126} \leq y \leq - \frac{41 - 2 \sqrt{4531}}{126}}$

Step 5

Determine the domain and range.

Domain: $(- \infty, \infty), {x | x \in ℝ}$

Range: $[- \frac{41 + 2 \sqrt{4531}}{126}, - \frac{41 - 2 \sqrt{4531}}{126}], {y | - \frac{41 + 2 \sqrt{4531}}{126} \leq y \leq - \frac{41 - 2 \sqrt{4531}}{126}}$

Step 6