Precalculus Examples

$\frac{5}{2 x^{2} + 3 x - 8}$

Step 1

Set the denominator in $\frac{5}{2 x^{2} + 3 x - 8}$ equal to $0$ to find where the expression is undefined.

$2 x^{2} + 3 x - 8 = 0$

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 $a = 2$ , $b = 3$ , and $c = - 8$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (2 \cdot - 8)}}{2 \cdot 2}$

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 $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 8}}{2 \cdot 2}$

Step 2.3.1.2

Multiply $- 4 \cdot 2 \cdot - 8$ .

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Step 2.3.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 8}}{2 \cdot 2}$

Step 2.3.1.2.2

Multiply $- 8$ by $- 8$ .

$x = \frac{- 3 \pm \sqrt{9 + 64}}{2 \cdot 2}$

Step 2.3.1.3

Add $9$ and $64$ .

$x = \frac{- 3 \pm \sqrt{73}}{2 \cdot 2}$

Step 2.3.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm \sqrt{73}}{4}$

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 $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 8}}{2 \cdot 2}$

Step 2.4.1.2

Multiply $- 4 \cdot 2 \cdot - 8$ .

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Step 2.4.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 8}}{2 \cdot 2}$

Step 2.4.1.2.2

Multiply $- 8$ by $- 8$ .

$x = \frac{- 3 \pm \sqrt{9 + 64}}{2 \cdot 2}$

Step 2.4.1.3

Add $9$ and $64$ .

$x = \frac{- 3 \pm \sqrt{73}}{2 \cdot 2}$

Step 2.4.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm \sqrt{73}}{4}$

Step 2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + \sqrt{73}}{4}$

Step 2.4.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + \sqrt{73}}{4}$

Step 2.4.5

Factor $- 1$ out of $\sqrt{73}$ .

$x = \frac{- 1 \cdot 3 - 1 (- \sqrt{73})}{4}$

Step 2.4.6

Factor $- 1$ out of $- 1 (3) - 1 (- \sqrt{73})$ .

$x = \frac{- 1 (3 - \sqrt{73})}{4}$

Step 2.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - \sqrt{73}}{4}$

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 $3$ to the power of $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 2 \cdot - 8}}{2 \cdot 2}$

Step 2.5.1.2

Multiply $- 4 \cdot 2 \cdot - 8$ .

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Step 2.5.1.2.1

Multiply $- 4$ by $2$ .

$x = \frac{- 3 \pm \sqrt{9 - 8 \cdot - 8}}{2 \cdot 2}$

Step 2.5.1.2.2

Multiply $- 8$ by $- 8$ .

$x = \frac{- 3 \pm \sqrt{9 + 64}}{2 \cdot 2}$

Step 2.5.1.3

Add $9$ and $64$ .

$x = \frac{- 3 \pm \sqrt{73}}{2 \cdot 2}$

Step 2.5.2

Multiply $2$ by $2$ .

$x = \frac{- 3 \pm \sqrt{73}}{4}$

Step 2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - \sqrt{73}}{4}$

Step 2.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - \sqrt{73}}{4}$

Step 2.5.5

Factor $- 1$ out of $- \sqrt{73}$ .

$x = \frac{- 1 \cdot 3 - (\sqrt{73})}{4}$

Step 2.5.6

Factor $- 1$ out of $- 1 (3) - (\sqrt{73})$ .

$x = \frac{- 1 (3 + \sqrt{73})}{4}$

Step 2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + \sqrt{73}}{4}$

Step 2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - \sqrt{73}}{4}, - \frac{3 + \sqrt{73}}{4}$

Step 3

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

Interval Notation:

$(- \infty, - \frac{3 + \sqrt{73}}{4}) \cup (- \frac{3 + \sqrt{73}}{4}, - \frac{3 - \sqrt{73}}{4}) \cup (- \frac{3 - \sqrt{73}}{4}, \infty)$

Set-Builder Notation:

${x | x \neq - \frac{3 - \sqrt{73}}{4}, - \frac{3 + \sqrt{73}}{4}}$

Step 4