Algebra Examples

\frac{x^{2} + 3 x - 10}{x^{2} - 4} \cdot \frac{x + 2}{x^{2} - 9}

$\frac{x^{2} + 3 x - 10}{x^{2} - 4} \cdot \frac{x + 2}{x^{2} - 9}$

Step 1

Factor

x^{2} + 3 x - 10

$x^{2} + 3 x - 10$ using the AC method.

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Step 1.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

- 10

$- 10$ and whose sum is

3

$3$ .

- 2, 5

$- 2, 5$

Step 1.2

Write the factored form using these integers.

\frac{(x - 2) (x + 5)}{x^{2} - 4} \cdot \frac{x + 2}{x^{2} - 9}

$\frac{(x - 2) (x + 5)}{x^{2} - 4} \cdot \frac{x + 2}{x^{2} - 9}$

\frac{(x - 2) (x + 5)}{x^{2} - 4} \cdot \frac{x + 2}{x^{2} - 9}

$\frac{(x - 2) (x + 5)}{x^{2} - 4} \cdot \frac{x + 2}{x^{2} - 9}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 2

$b = 2$ .

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

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

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

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

Step 3

Simplify the denominator.

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Step 3.1

Rewrite

9

$9$ as

3^{2}

$3^{2}$ .

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

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

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula,

a^{2} - b^{2} = (a + b) (a - b)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = x

$a = x$ and

b = 3

$b = 3$ .

\frac{(x - 2) (x + 5)}{(x + 2) (x - 2)} \cdot \frac{x + 2}{(x + 3) (x - 3)}

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

\frac{(x - 2) (x + 5)}{(x + 2) (x - 2)} \cdot \frac{x + 2}{(x + 3) (x - 3)}

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of

x + 2

$x + 2$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

$\frac{(x - 2) (x + 5)}{x - 2} \cdot \frac{1}{(x + 3) (x - 3)}$

Step 4.2

Multiply

$\frac{(x - 2) (x + 5)}{x - 2}$ by

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

$\frac{(x - 2) (x + 5)}{(x - 2) ((x + 3) (x - 3))}$

Step 4.3

Cancel the common factor of

$x - 2$ .

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Step 4.3.1

Cancel the common factor.

$\frac{(x - 2) (x + 5)}{(x - 2) ((x + 3) (x - 3))}$

Step 4.3.2

Rewrite the expression.

$\frac{x + 5}{(x + 3) (x - 3)}$