Pre-Algebra Examples

$\frac{3 x - 5}{x^{2} + 3 x + 2} + \frac{x}{x + 2}$

Step 1

Factor $x^{2} + 3 x + 2$ 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 $2$ and whose sum is $3$ .

$1, 2$

Step 1.2

Write the factored form using these integers.

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

Step 2

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Reorder the factors of $(x + 2) (x + 1)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Multiply $x$ by $x$ .

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

Step 5.3

Multiply $x$ by $1$ .

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

Step 5.4

Add $3 x$ and $x$ .

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

Step 5.5

Reorder terms.

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

Step 5.6

Factor $x^{2} + 4 x - 5$ using the AC method.

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Step 5.6.1

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 $- 5$ and whose sum is $4$ .

$- 1, 5$

Step 5.6.2

Write the factored form using these integers.

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