Precalculus Examples

$\frac{x + h}{x + h + 2} - \frac{x}{x + 2}$

Step 1

To write $\frac{x + h}{x + h + 2}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

$\frac{x + h}{x + h + 2} \cdot \frac{x + 2}{x + 2} - \frac{x}{x + 2}$

Step 2

To write $- \frac{x}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x + h + 2}{x + h + 2}$ .

$\frac{x + h}{x + h + 2} \cdot \frac{x + 2}{x + 2} - \frac{x}{x + 2} \cdot \frac{x + h + 2}{x + h + 2}$

Step 3

Write each expression with a common denominator of $(x + h + 2) (x + 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{x + h}{x + h + 2}$ by $\frac{x + 2}{x + 2}$ .

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

Step 3.2

Multiply $\frac{x}{x + 2}$ by $\frac{x + h + 2}{x + h + 2}$ .

$\frac{(x + h) (x + 2)}{(x + h + 2) (x + 2)} - \frac{x (x + h + 2)}{(x + 2) (x + h + 2)}$

Step 3.3

Reorder the factors of $(x + h + 2) (x + 2)$ .

$\frac{(x + h) (x + 2)}{(x + 2) (x + h + 2)} - \frac{x (x + h + 2)}{(x + 2) (x + h + 2)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x + h) (x + 2) - x (x + h + 2)}{(x + 2) (x + h + 2)}$

Step 5

Simplify the numerator.

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Step 5.1

Expand $(x + h) (x + 2)$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

$\frac{x (x + 2) + h (x + 2) - x (x + h + 2)}{(x + 2) (x + h + 2)}$

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot 2 + h x + h \cdot 2 - x (x + h + 2)}{(x + 2) (x + h + 2)}$

Step 5.2.2

Move $2$ to the left of $x$ .

$\frac{x^{2} + 2 \cdot x + h x + h \cdot 2 - x (x + h + 2)}{(x + 2) (x + h + 2)}$

Step 5.2.3

Move $2$ to the left of $h$ .

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

Step 5.3

Apply the distributive property.

$\frac{x^{2} + 2 x + h x + 2 h - x \cdot x - x h - x \cdot 2}{(x + 2) (x + h + 2)}$

Step 5.4

Simplify.

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Step 5.4.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.4.1.1

Move $x$ .

$\frac{x^{2} + 2 x + h x + 2 h - (x \cdot x) - x h - x \cdot 2}{(x + 2) (x + h + 2)}$

Step 5.4.1.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 2 x + h x + 2 h - x^{2} - x h - x \cdot 2}{(x + 2) (x + h + 2)}$

Step 5.4.2

Multiply $2$ by $- 1$ .

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

Step 5.5

Subtract $x^{2}$ from $x^{2}$ .

$\frac{2 x + h x + 2 h + 0 - x h - 2 x}{(x + 2) (x + h + 2)}$

Step 5.6

Add $2 x$ and $0$ .

$\frac{h x + 2 h + 2 x - x h - 2 x}{(x + 2) (x + h + 2)}$

Step 5.7

Subtract $x h$ from $h x$ .

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Step 5.7.1

Move $x$ .

$\frac{2 h + 2 x + h x - h x - 2 x}{(x + 2) (x + h + 2)}$

Step 5.7.2

Subtract $h x$ from $h x$ .

$\frac{2 h + 2 x + 0 - 2 x}{(x + 2) (x + h + 2)}$

Step 5.8

Add $2 h$ and $0$ .

$\frac{2 h + 2 x - 2 x}{(x + 2) (x + h + 2)}$

Step 5.9

Subtract $2 x$ from $2 x$ .

$\frac{2 h + 0}{(x + 2) (x + h + 2)}$

Step 5.10

Add $2 h$ and $0$ .

$\frac{2 h}{(x + 2) (x + h + 2)}$