Algebra Examples

$\frac{x^{2} - 11 x + 30}{x^{2} - 4} \times \frac{x - 2}{x^{2} + 3 x - 40}$

Step 1

Factor $x^{2} - 11 x + 30$ 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 $30$ and whose sum is $- 11$ .

$- 6, - 5$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 6) (x - 5)}{x^{2} - 4} \cdot \frac{x - 2}{x^{2} + 3 x - 40}$

Step 2

Simplify the denominator.

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Step 2.1

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

$\frac{(x - 6) (x - 5)}{x^{2} - 2^{2}} \cdot \frac{x - 2}{x^{2} + 3 x - 40}$

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 $b = 2$ .

$\frac{(x - 6) (x - 5)}{(x + 2) (x - 2)} \cdot \frac{x - 2}{x^{2} + 3 x - 40}$

Step 3

Factor $x^{2} + 3 x - 40$ 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 $- 40$ and whose sum is $3$ .

$- 5, 8$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 6) (x - 5)}{(x + 2) (x - 2)} \cdot \frac{x - 2}{(x - 5) (x + 8)}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 5$ .

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Step 4.1.1

Factor $x - 5$ out of $(x - 6) (x - 5)$ .

$\frac{(x - 5) (x - 6)}{(x + 2) (x - 2)} \cdot \frac{x - 2}{(x - 5) (x + 8)}$

Step 4.1.2

Cancel the common factor.

$\frac{(x - 5) (x - 6)}{(x + 2) (x - 2)} \cdot \frac{x - 2}{(x - 5) (x + 8)}$

Step 4.1.3

Rewrite the expression.

$\frac{x - 6}{(x + 2) (x - 2)} \cdot \frac{x - 2}{x + 8}$

Step 4.2

Cancel the common factor of $x - 2$ .

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Step 4.2.1

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

$\frac{x - 6}{(x - 2) (x + 2)} \cdot \frac{x - 2}{x + 8}$

Step 4.2.2

Cancel the common factor.

$\frac{x - 6}{(x - 2) (x + 2)} \cdot \frac{x - 2}{x + 8}$

Step 4.2.3

Rewrite the expression.

$\frac{x - 6}{x + 2} \cdot \frac{1}{x + 8}$

Step 4.3

Multiply $\frac{x - 6}{x + 2}$ by $\frac{1}{x + 8}$ .

$\frac{x - 6}{(x + 2) (x + 8)}$