Algebra Examples

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

Step 1

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

$- 3, - 2$

Step 1.2

Write the factored form using these integers.

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

Step 2

Simplify the denominator.

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Step 2.1

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

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

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 $b = 2$ .

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

Step 3

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

$1, 2$

Step 3.2

Write the factored form using these integers.

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

Step 4

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

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Step 4.1

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 $- 3$ and whose sum is $- 2$ .

$- 3, 1$

Step 4.2

Write the factored form using these integers.

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

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x - 3$ .

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

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

Step 5.2

Cancel the common factor of $x + 2$ .

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 5.3

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

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

Step 5.4

Cancel the common factor of $x - 2$ .

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Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Rewrite the expression.

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

Step 5.5

Cancel the common factor of $x + 1$ .

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Step 5.5.1

Cancel the common factor.

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

Step 5.5.2

Rewrite the expression.

$1$