Algebra Examples

$\frac{3 x}{2 x + 2} \cdot \frac{x^{2} - 1}{x^{2} + 6 x - 7}$

Step 1

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

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Step 1.1

Factor $2$ out of $2 x$ .

$\frac{3 x}{2 (x) + 2} \cdot \frac{x^{2} - 1}{x^{2} + 6 x - 7}$

Step 1.2

Factor $2$ out of $2$ .

$\frac{3 x}{2 (x) + 2 (1)} \cdot \frac{x^{2} - 1}{x^{2} + 6 x - 7}$

Step 1.3

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

$\frac{3 x}{2 (x + 1)} \cdot \frac{x^{2} - 1}{x^{2} + 6 x - 7}$

Step 2

Simplify the numerator.

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Step 2.1

Rewrite $1$ as $1^{2}$ .

$\frac{3 x}{2 (x + 1)} \cdot \frac{x^{2} - 1^{2}}{x^{2} + 6 x - 7}$

Step 2.2

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 $b = 1$ .

$\frac{3 x}{2 (x + 1)} \cdot \frac{(x + 1) (x - 1)}{x^{2} + 6 x - 7}$

Step 3

Factor $x^{2} + 6 x - 7$ using the AC method.

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Step 3.1

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 $- 7$ and whose sum is $6$ .

$- 1, 7$

Step 3.2

Write the factored form using these integers.

$\frac{3 x}{2 (x + 1)} \cdot \frac{(x + 1) (x - 1)}{(x - 1) (x + 7)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 1$ .

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Step 4.1.1

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

$\frac{3 x}{(x + 1) \cdot 2} \cdot \frac{(x + 1) (x - 1)}{(x - 1) (x + 7)}$

Step 4.1.2

Cancel the common factor.

$\frac{3 x}{(x + 1) \cdot 2} \cdot \frac{(x + 1) (x - 1)}{(x - 1) (x + 7)}$

Step 4.1.3

Rewrite the expression.

$\frac{3 x}{2} \cdot \frac{x - 1}{(x - 1) (x + 7)}$

Step 4.2

Multiply $\frac{3 x}{2}$ by $\frac{x - 1}{(x - 1) (x + 7)}$ .

$\frac{3 x (x - 1)}{2 ((x - 1) (x + 7))}$

Step 4.3

Cancel the common factor of $x - 1$ .

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Step 4.3.1

Cancel the common factor.

$\frac{3 x (x - 1)}{2 ((x - 1) (x + 7))}$

Step 4.3.2

Rewrite the expression.

$\frac{3 x}{2 (x + 7)}$