Calculus Examples

$\frac{\frac{1}{x + 2} + 3}{\frac{4}{x} - x}$

Step 1

Multiply the numerator and denominator of the fraction by $(x + 2) x$ .

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Step 1.1

Multiply $\frac{\frac{1}{x + 2} + 3}{\frac{4}{x} - x}$ by $\frac{(x + 2) x}{(x + 2) x}$ .

$\frac{(x + 2) x}{(x + 2) x} \cdot \frac{\frac{1}{x + 2} + 3}{\frac{4}{x} - x}$

Step 1.2

Combine.

$\frac{(x + 2) x (\frac{1}{x + 2} + 3)}{(x + 2) x (\frac{4}{x} - x)}$

Step 2

Apply the distributive property.

$\frac{(x + 2) x \frac{1}{x + 2} + (x + 2) x \cdot 3}{(x + 2) x \frac{4}{x} + (x + 2) x (- x)}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x + 2$ .

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Step 3.1.1

Factor $x + 2$ out of $(x + 2) x$ .

$\frac{(x + 2) (x) \frac{1}{x + 2} + (x + 2) x \cdot 3}{(x + 2) x \frac{4}{x} + (x + 2) x (- x)}$

Step 3.1.2

Cancel the common factor.

$\frac{(x + 2) x \frac{1}{x + 2} + (x + 2) x \cdot 3}{(x + 2) x \frac{4}{x} + (x + 2) x (- x)}$

Step 3.1.3

Rewrite the expression.

$\frac{x + (x + 2) x \cdot 3}{(x + 2) x \frac{4}{x} + (x + 2) x (- x)}$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Factor $x$ out of $(x + 2) x$ .

$\frac{x + (x + 2) x \cdot 3}{x (x + 2) \frac{4}{x} + (x + 2) x (- x)}$

Step 3.2.2

Cancel the common factor.

$\frac{x + (x + 2) x \cdot 3}{x (x + 2) \frac{4}{x} + (x + 2) x (- x)}$

Step 3.2.3

Rewrite the expression.

$\frac{x + (x + 2) x \cdot 3}{(x + 2) \cdot 4 + (x + 2) x (- x)}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $x$ out of $x + (x + 2) x \cdot 3$ .

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Step 4.1.1

Raise $x$ to the power of $1$ .

$\frac{x^{1} + (x + 2) x \cdot 3}{(x + 2) \cdot 4 + (x + 2) x (- x)}$

Step 4.1.2

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot 1 + (x + 2) x \cdot 3}{(x + 2) \cdot 4 + (x + 2) x (- x)}$

Step 4.1.3

Factor $x$ out of $(x + 2) x \cdot 3$ .

$\frac{x \cdot 1 + x ((x + 2) \cdot 3)}{(x + 2) \cdot 4 + (x + 2) x (- x)}$

Step 4.1.4

Factor $x$ out of $x \cdot 1 + x ((x + 2) \cdot 3)$ .

$\frac{x (1 + (x + 2) \cdot 3)}{(x + 2) \cdot 4 + (x + 2) x (- x)}$

Step 4.2

Apply the distributive property.

$\frac{x (1 + x \cdot 3 + 2 \cdot 3)}{(x + 2) \cdot 4 + (x + 2) x (- x)}$

Step 4.3

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

$\frac{x (1 + 3 \cdot x + 2 \cdot 3)}{(x + 2) \cdot 4 + (x + 2) x (- x)}$

Step 4.4

Multiply $2$ by $3$ .

$\frac{x (1 + 3 \cdot x + 6)}{(x + 2) \cdot 4 + (x + 2) x (- x)}$

Step 4.5

Add $1$ and $6$ .

$\frac{x (3 x + 7)}{(x + 2) \cdot 4 + (x + 2) x (- x)}$

Step 5

Simplify the denominator.

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Step 5.1

Factor $x + 2$ out of $(x + 2) \cdot 4 + (x + 2) x (- x)$ .

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Step 5.1.1

Factor $x + 2$ out of $(x + 2) \cdot 4$ .

$\frac{x (3 x + 7)}{(x + 2) (4) + (x + 2) x (- x)}$

Step 5.1.2

Factor $x + 2$ out of $(x + 2) x (- x)$ .

$\frac{x (3 x + 7)}{(x + 2) (4) + (x + 2) (x (- x))}$

Step 5.1.3

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

$\frac{x (3 x + 7)}{(x + 2) (4 + x (- x))}$

Step 5.2

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

$\frac{x (3 x + 7)}{(x + 2) (2^{2} + x (- x))}$

Step 5.3

Rewrite $x (x)$ as $x^{2}$ .

$\frac{x (3 x + 7)}{(x + 2) (2^{2} - x^{2})}$

Step 5.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2$ and $b = x$ .

$\frac{x (3 x + 7)}{(x + 2) (2 + x) (2 - x)}$

Step 5.5

Combine exponents.

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Step 5.5.1

Reorder terms.

$\frac{x (3 x + 7)}{(x + 2) (x + 2) (2 - x)}$

Step 5.5.2

Raise $x + 2$ to the power of $1$ .

$\frac{x (3 x + 7)}{{(x + 2)}^{1} (x + 2) (2 - x)}$

Step 5.5.3

Raise $x + 2$ to the power of $1$ .

$\frac{x (3 x + 7)}{{(x + 2)}^{1} {(x + 2)}^{1} (2 - x)}$

Step 5.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{x (3 x + 7)}{{(x + 2)}^{1 + 1} (2 - x)}$

Step 5.5.5

Add $1$ and $1$ .

$\frac{x (3 x + 7)}{{(x + 2)}^{2} (2 - x)}$