Finite Math Examples

$\frac{7 x^{2} - 9 x}{x^{2} - 4 x - 7}$

Step 1

Set the denominator in $\frac{7 x^{2} - 9 x}{x^{2} - 4 x - 7}$ equal to $0$ to find where the expression is undefined.

$x^{2} - 4 x - 7 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Use the quadratic formula to find the solutions.

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

Step 2.2

Substitute the values $a = 1$ , $b = - 4$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (1 \cdot - 7)}}{2 \cdot 1}$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the numerator.

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Step 2.3.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Step 2.3.1.2

Multiply $- 4 \cdot 1 \cdot - 7$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 7}}{2 \cdot 1}$

Step 2.3.1.2.2

Multiply $- 4$ by $- 7$ .

$x = \frac{4 \pm \sqrt{16 + 28}}{2 \cdot 1}$

Step 2.3.1.3

Add $16$ and $28$ .

$x = \frac{4 \pm \sqrt{44}}{2 \cdot 1}$

Step 2.3.1.4

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 2.3.1.4.1

Factor $4$ out of $44$ .

$x = \frac{4 \pm \sqrt{4 (11)}}{2 \cdot 1}$

Step 2.3.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Step 2.3.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{11}}{2 \cdot 1}$

Step 2.3.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{11}}{2}$

Step 2.3.3

Simplify $\frac{4 \pm 2 \sqrt{11}}{2}$ .

$x = 2 \pm \sqrt{11}$

Step 2.4

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

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Step 2.4.1.2

Multiply $- 4 \cdot 1 \cdot - 7$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 7}}{2 \cdot 1}$

Step 2.4.1.2.2

Multiply $- 4$ by $- 7$ .

$x = \frac{4 \pm \sqrt{16 + 28}}{2 \cdot 1}$

Step 2.4.1.3

Add $16$ and $28$ .

$x = \frac{4 \pm \sqrt{44}}{2 \cdot 1}$

Step 2.4.1.4

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 2.4.1.4.1

Factor $4$ out of $44$ .

$x = \frac{4 \pm \sqrt{4 (11)}}{2 \cdot 1}$

Step 2.4.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Step 2.4.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{11}}{2 \cdot 1}$

Step 2.4.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{11}}{2}$

Step 2.4.3

Simplify $\frac{4 \pm 2 \sqrt{11}}{2}$ .

$x = 2 \pm \sqrt{11}$

Step 2.4.4

Change the $\pm$ to $+$ .

$x = 2 + \sqrt{11}$

Step 2.5

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

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $- 4$ to the power of $2$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot 1 \cdot - 7}}{2 \cdot 1}$

Step 2.5.1.2

Multiply $- 4 \cdot 1 \cdot - 7$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 7}}{2 \cdot 1}$

Step 2.5.1.2.2

Multiply $- 4$ by $- 7$ .

$x = \frac{4 \pm \sqrt{16 + 28}}{2 \cdot 1}$

Step 2.5.1.3

Add $16$ and $28$ .

$x = \frac{4 \pm \sqrt{44}}{2 \cdot 1}$

Step 2.5.1.4

Rewrite $44$ as $2^{2} \cdot 11$ .

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Step 2.5.1.4.1

Factor $4$ out of $44$ .

$x = \frac{4 \pm \sqrt{4 (11)}}{2 \cdot 1}$

Step 2.5.1.4.2

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

$x = \frac{4 \pm \sqrt{2^{2} \cdot 11}}{2 \cdot 1}$

Step 2.5.1.5

Pull terms out from under the radical.

$x = \frac{4 \pm 2 \sqrt{11}}{2 \cdot 1}$

Step 2.5.2

Multiply $2$ by $1$ .

$x = \frac{4 \pm 2 \sqrt{11}}{2}$

Step 2.5.3

Simplify $\frac{4 \pm 2 \sqrt{11}}{2}$ .

$x = 2 \pm \sqrt{11}$

Step 2.5.4

Change the $\pm$ to $-$ .

$x = 2 - \sqrt{11}$

Step 2.6

The final answer is the combination of both solutions.

$x = 2 + \sqrt{11}, 2 - \sqrt{11}$

Step 3

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

Interval Notation:

$(- \infty, 2 - \sqrt{11}) \cup (2 - \sqrt{11}, 2 + \sqrt{11}) \cup (2 + \sqrt{11}, \infty)$

Set-Builder Notation:

${x | x \neq 2 - \sqrt{11}, 2 + \sqrt{11}}$

Step 4