Algebra Examples

$\frac{5}{x + 5} - \frac{2 x + 5}{x^{2} + 9 x + 20}$

Step 1

Factor $x^{2} + 9 x + 20$ using the AC method.

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Step 1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $20$ and whose sum is $9$ .

$4, 5$

Step 1.2

Write the factored form using these integers.

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

Step 2

To write $\frac{5}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

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

Step 3

Write each expression with a common denominator of $(x + 5) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{5}{x + 5}$ by $\frac{x + 4}{x + 4}$ .

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

Step 3.2

Reorder the factors of $(x + 5) (x + 4)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $5$ by $4$ .

$\frac{5 x + 20 - (2 x + 5)}{(x + 4) (x + 5)}$

Step 5.3

Apply the distributive property.

$\frac{5 x + 20 - (2 x) - 1 \cdot 5}{(x + 4) (x + 5)}$

Step 5.4

Multiply $2$ by $- 1$ .

$\frac{5 x + 20 - 2 x - 1 \cdot 5}{(x + 4) (x + 5)}$

Step 5.5

Multiply $- 1$ by $5$ .

$\frac{5 x + 20 - 2 x - 5}{(x + 4) (x + 5)}$

Step 5.6

Subtract $2 x$ from $5 x$ .

$\frac{3 x + 20 - 5}{(x + 4) (x + 5)}$

Step 5.7

Subtract $5$ from $20$ .

$\frac{3 x + 15}{(x + 4) (x + 5)}$

Step 5.8

Factor $3$ out of $3 x + 15$ .

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Step 5.8.1

Factor $3$ out of $3 x$ .

$\frac{3 (x) + 15}{(x + 4) (x + 5)}$

Step 5.8.2

Factor $3$ out of $15$ .

$\frac{3 x + 3 \cdot 5}{(x + 4) (x + 5)}$

Step 5.8.3

Factor $3$ out of $3 x + 3 \cdot 5$ .

$\frac{3 (x + 5)}{(x + 4) (x + 5)}$

Step 6

Cancel the common factor of $x + 5$ .

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Step 6.1

Cancel the common factor.

$\frac{3 (x + 5)}{(x + 4) (x + 5)}$

Step 6.2

Rewrite the expression.

$\frac{3}{x + 4}$