Calculus Examples

$4 + \frac{5}{x^{2} + 2}$

Step 1

Set $4 + \frac{5}{x^{2} + 2}$ equal to $0$ .

$4 + \frac{5}{x^{2} + 2} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $4$ from both sides of the equation.

$\frac{5}{x^{2} + 2} = - 4$

Step 2.2

Find the LCD of the terms in the equation.

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Step 2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x^{2} + 2, 1$

Step 2.2.2

Remove parentheses.

$x^{2} + 2, 1$

Step 2.2.3

The LCM of one and any expression is the expression.

$x^{2} + 2$

Step 2.3

Multiply each term in $\frac{5}{x^{2} + 2} = - 4$ by $x^{2} + 2$ to eliminate the fractions.

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Step 2.3.1

Multiply each term in $\frac{5}{x^{2} + 2} = - 4$ by $x^{2} + 2$ .

$\frac{5}{x^{2} + 2} (x^{2} + 2) = - 4 (x^{2} + 2)$

Step 2.3.2

Simplify the left side.

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Step 2.3.2.1

Cancel the common factor of $x^{2} + 2$ .

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Step 2.3.2.1.1

Cancel the common factor.

$\frac{5}{x^{2} + 2} (x^{2} + 2) = - 4 (x^{2} + 2)$

Step 2.3.2.1.2

Rewrite the expression.

$5 = - 4 (x^{2} + 2)$

Step 2.3.3

Simplify the right side.

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Step 2.3.3.1

Apply the distributive property.

$5 = - 4 x^{2} - 4 \cdot 2$

Step 2.3.3.2

Multiply $- 4$ by $2$ .

$5 = - 4 x^{2} - 8$

Step 2.4

Solve the equation.

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Step 2.4.1

Rewrite the equation as $- 4 x^{2} - 8 = 5$ .

$- 4 x^{2} - 8 = 5$

Step 2.4.2

Move all terms not containing $x$ to the right side of the equation.

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Step 2.4.2.1

Add $8$ to both sides of the equation.

$- 4 x^{2} = 5 + 8$

Step 2.4.2.2

Add $5$ and $8$ .

$- 4 x^{2} = 13$

Step 2.4.3

Divide each term in $- 4 x^{2} = 13$ by $- 4$ and simplify.

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Step 2.4.3.1

Divide each term in $- 4 x^{2} = 13$ by $- 4$ .

$\frac{- 4 x^{2}}{- 4} = \frac{13}{- 4}$

Step 2.4.3.2

Simplify the left side.

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Step 2.4.3.2.1

Cancel the common factor of $- 4$ .

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Step 2.4.3.2.1.1

Cancel the common factor.

$\frac{- 4 x^{2}}{- 4} = \frac{13}{- 4}$

Step 2.4.3.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{13}{- 4}$

Step 2.4.3.3

Simplify the right side.

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Step 2.4.3.3.1

Move the negative in front of the fraction.

$x^{2} = - \frac{13}{4}$

Step 2.4.4

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

$x = \pm \sqrt{- \frac{13}{4}}$

Step 2.4.5

Simplify $\pm \sqrt{- \frac{13}{4}}$ .

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Step 2.4.5.1

Rewrite $- \frac{13}{4}$ as ${(\frac{i}{2})}^{2} \cdot 13$ .

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Step 2.4.5.1.1

Rewrite $- 1$ as $i^{2}$ .

$x = \pm \sqrt{i^{2} \frac{13}{4}}$

Step 2.4.5.1.2

Factor the perfect power $1^{2}$ out of $13$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 13}{4}}$

Step 2.4.5.1.3

Factor the perfect power $2^{2}$ out of $4$ .

$x = \pm \sqrt{i^{2} \frac{1^{2} \cdot 13}{2^{2} \cdot 1}}$

Step 2.4.5.1.4

Rearrange the fraction $\frac{1^{2} \cdot 13}{2^{2} \cdot 1}$ .

$x = \pm \sqrt{i^{2} ({(\frac{1}{2})}^{2} \cdot 13)}$

Step 2.4.5.1.5

Rewrite $i^{2} {(\frac{1}{2})}^{2}$ as ${(\frac{i}{2})}^{2}$ .

$x = \pm \sqrt{{(\frac{i}{2})}^{2} \cdot 13}$

Step 2.4.5.2

Pull terms out from under the radical.

$x = \pm \frac{i}{2} \sqrt{13}$

Step 2.4.5.3

Combine $\frac{i}{2}$ and $\sqrt{13}$ .

$x = \pm \frac{i \sqrt{13}}{2}$

Step 2.4.6

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

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Step 2.4.6.1

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

$x = \frac{i \sqrt{13}}{2}$

Step 2.4.6.2

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

$x = - \frac{i \sqrt{13}}{2}$

Step 2.4.6.3

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

$x = \frac{i \sqrt{13}}{2}, - \frac{i \sqrt{13}}{2}$

Step 3