Pre-Algebra Examples

$\frac{\frac{x + 2}{x^{2} - 3 x + 2}}{\frac{x + 3}{4 x - 4}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x + 2}{x^{2} - 3 x + 2} \cdot \frac{4 x - 4}{x + 3}$

Step 2

Factor $x^{2} - 3 x + 2$ using the AC method.

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Step 2.1

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 $- 3$ .

$- 2, - 1$

Step 2.2

Write the factored form using these integers.

$\frac{x + 2}{(x - 2) (x - 1)} \cdot \frac{4 x - 4}{x + 3}$

Step 3

Factor $4$ out of $4 x - 4$ .

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Step 3.1

Factor $4$ out of $4 x$ .

$\frac{x + 2}{(x - 2) (x - 1)} \cdot \frac{4 (x) - 4}{x + 3}$

Step 3.2

Factor $4$ out of $- 4$ .

$\frac{x + 2}{(x - 2) (x - 1)} \cdot \frac{4 (x) + 4 (- 1)}{x + 3}$

Step 3.3

Factor $4$ out of $4 (x) + 4 (- 1)$ .

$\frac{x + 2}{(x - 2) (x - 1)} \cdot \frac{4 (x - 1)}{x + 3}$

Step 4

Cancel the common factor of $x - 1$ .

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Step 4.1

Factor $x - 1$ out of $(x - 2) (x - 1)$ .

$\frac{x + 2}{(x - 1) (x - 2)} \cdot \frac{4 (x - 1)}{x + 3}$

Step 4.2

Factor $x - 1$ out of $4 (x - 1)$ .

$\frac{x + 2}{(x - 1) (x - 2)} \cdot \frac{(x - 1) \cdot 4}{x + 3}$

Step 4.3

Cancel the common factor.

$\frac{x + 2}{(x - 1) (x - 2)} \cdot \frac{(x - 1) \cdot 4}{x + 3}$

Step 4.4

Rewrite the expression.

$\frac{x + 2}{x - 2} \cdot \frac{4}{x + 3}$

Step 5

Multiply $\frac{x + 2}{x - 2}$ by $\frac{4}{x + 3}$ .

$\frac{(x + 2) \cdot 4}{(x - 2) (x + 3)}$

Step 6

Move $4$ to the left of $x + 2$ .

$\frac{4 (x + 2)}{(x - 2) (x + 3)}$

Step 7

Apply the distributive property.

$\frac{4 x + 4 \cdot 2}{(x - 2) (x + 3)}$

Step 8

Multiply $4$ by $2$ .

$\frac{4 x + 8}{(x - 2) (x + 3)}$

Step 9

Split the fraction $\frac{4 x + 8}{(x - 2) (x + 3)}$ into two fractions.

$\frac{4 x}{(x - 2) (x + 3)} + \frac{8}{(x - 2) (x + 3)}$