Pre-Algebra Examples

$\frac{1}{x + 2} + \frac{1}{x + 3} + \frac{1}{x^{2} + 5 x + 6}$

Step 1

Factor $x^{2} + 5 x + 6$ using the AC method.

Tap for more steps...

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 $6$ and whose sum is $5$ .

$2, 3$

Step 1.2

Write the factored form using these integers.

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

Step 2

To write $\frac{1}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

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

Step 3

To write $\frac{1}{x + 3}$ as a fraction with a common denominator, multiply by $\frac{x + 2}{x + 2}$ .

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

Step 4

Write each expression with a common denominator of $(x + 2) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.1

Multiply $\frac{1}{x + 2}$ by $\frac{x + 3}{x + 3}$ .

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

Step 4.2

Multiply $\frac{1}{x + 3}$ by $\frac{x + 2}{x + 2}$ .

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

Step 4.3

Reorder the factors of $(x + 3) (x + 2)$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Add $x$ and $x$ .

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

Step 8

Simplify by adding numbers.

Tap for more steps...

Step 8.1

Add $3$ and $2$ .

$\frac{2 x + 5 + 1}{(x + 2) (x + 3)}$

Step 8.2

Add $5$ and $1$ .

$\frac{2 x + 6}{(x + 2) (x + 3)}$

Step 9

Factor $2$ out of $2 x + 6$ .

Tap for more steps...

Step 9.1

Factor $2$ out of $2 x$ .

$\frac{2 (x) + 6}{(x + 2) (x + 3)}$

Step 9.2

Factor $2$ out of $6$ .

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

Step 9.3

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

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

Step 10

Cancel the common factor of $x + 3$ .

Tap for more steps...

Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

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