Algebra Examples

$\frac{x^{2} - 36}{x^{2} - 5 x - 6} \cdot \frac{3 x + 3}{9 x^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $36$ as $6^{2}$ .

$\frac{x^{2} - 6^{2}}{x^{2} - 5 x - 6} \cdot \frac{3 x + 3}{9 x^{2}}$

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 $b = 6$ .

$\frac{(x + 6) (x - 6)}{x^{2} - 5 x - 6} \cdot \frac{3 x + 3}{9 x^{2}}$

Step 2

Factor $x^{2} - 5 x - 6$ 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 $- 6$ and whose sum is $- 5$ .

$- 6, 1$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{3 x + 3}{9 x^{2}}$

Step 3

Simplify terms.

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Step 3.1

Factor $3$ out of $3 x + 3$ .

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Step 3.1.1

Factor $3$ out of $3 x$ .

$\frac{(x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{3 (x) + 3}{9 x^{2}}$

Step 3.1.2

Factor $3$ out of $3$ .

$\frac{(x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{3 (x) + 3 (1)}{9 x^{2}}$

Step 3.1.3

Factor $3$ out of $3 (x) + 3 (1)$ .

$\frac{(x + 6) (x - 6)}{(x - 6) (x + 1)} \cdot \frac{3 (x + 1)}{9 x^{2}}$

Step 3.2

Combine.

$\frac{(x + 6) (x - 6) (3 (x + 1))}{(x - 6) (x + 1) (9 x^{2})}$

Step 3.3

Cancel the common factor of $x - 6$ .

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Step 3.3.1

Cancel the common factor.

$\frac{(x + 6) (x - 6) (3 (x + 1))}{(x - 6) (x + 1) (9 x^{2})}$

Step 3.3.2

Rewrite the expression.

$\frac{(x + 6) (3 (x + 1))}{(x + 1) (9 x^{2})}$

Step 3.4

Cancel the common factor of $3$ and $9$ .

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Step 3.4.1

Factor $3$ out of $(x + 6) (3 (x + 1))$ .

$\frac{3 ((x + 6) (x + 1))}{(x + 1) (9 x^{2})}$

Step 3.4.2

Cancel the common factors.

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Step 3.4.2.1

Factor $3$ out of $(x + 1) (9 x^{2})$ .

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

Step 3.4.2.2

Cancel the common factor.

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

Step 3.4.2.3

Rewrite the expression.

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

Step 3.5

Cancel the common factor of $x + 1$ .

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Step 3.5.1

Cancel the common factor.

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

Step 3.5.2

Rewrite the expression.

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