Algebra Examples

$\frac{x + 5}{x + 2} - \frac{x + 1}{x^{2} + 2 x}$

Step 1

Factor $x$ out of $x^{2} + 2 x$ .

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Step 1.1

Factor $x$ out of $x^{2}$ .

$\frac{x + 5}{x + 2} - \frac{x + 1}{x \cdot x + 2 x}$

Step 1.2

Factor $x$ out of $2 x$ .

$\frac{x + 5}{x + 2} - \frac{x + 1}{x \cdot x + x \cdot 2}$

Step 1.3

Factor $x$ out of $x \cdot x + x \cdot 2$ .

$\frac{x + 5}{x + 2} - \frac{x + 1}{x (x + 2)}$

Step 2

To write $\frac{x + 5}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{x + 5}{x + 2} \cdot \frac{x}{x} - \frac{x + 1}{x (x + 2)}$

Step 3

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

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Step 3.1

Multiply $\frac{x + 5}{x + 2}$ by $\frac{x}{x}$ .

$\frac{(x + 5) x}{(x + 2) x} - \frac{x + 1}{x (x + 2)}$

Step 3.2

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

$\frac{(x + 5) x}{x (x + 2)} - \frac{x + 1}{x (x + 2)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(x + 5) x - (x + 1)}{x (x + 2)}$

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

$\frac{x \cdot x + 5 x - (x + 1)}{x (x + 2)}$

Step 5.2

Multiply $x$ by $x$ .

$\frac{x^{2} + 5 x - (x + 1)}{x (x + 2)}$

Step 5.3

Apply the distributive property.

$\frac{x^{2} + 5 x - x - 1 \cdot 1}{x (x + 2)}$

Step 5.4

Multiply $- 1$ by $1$ .

$\frac{x^{2} + 5 x - x - 1}{x (x + 2)}$

Step 5.5

Subtract $x$ from $5 x$ .

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