Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $4$ as $2^{2}$ .

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

Step 1.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 2$ .

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

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 $3$ .

$1, 2$

Step 2.2

Write the factored form using these integers.

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

Step 3

Simplify terms.

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Step 3.1

Cancel the common factor of $x - 2$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

Cancel the common factor of $x + 2$ .

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Step 3.4.1

Cancel the common factor.

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

Step 3.4.2

Rewrite the expression.

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

Step 3.5

Move $3$ to the left of $x$ .

$\frac{3 x}{x + 1}$