Algebra Examples

$\frac{x^{2} - 2 x - 24}{x^{2} + 7 x + 12} \cdot \frac{x^{2} - 1}{x - 6}$

Step 1

Factor $x^{2} - 2 x - 24$ 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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

Step 1.2

Write the factored form using these integers.

$\frac{(x - 6) (x + 4)}{x^{2} + 7 x + 12} \cdot \frac{x^{2} - 1}{x - 6}$

Step 2

Factor $x^{2} + 7 x + 12$ 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 $12$ and whose sum is $7$ .

$3, 4$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 6) (x + 4)}{(x + 3) (x + 4)} \cdot \frac{x^{2} - 1}{x - 6}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{(x - 6) (x + 4)}{(x + 3) (x + 4)} \cdot \frac{x^{2} - 1^{2}}{x - 6}$

Step 3.2

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 $b = 1$ .

$\frac{(x - 6) (x + 4)}{(x + 3) (x + 4)} \cdot \frac{(x + 1) (x - 1)}{x - 6}$

Step 4

Reduce the expression by cancelling the common factors.

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Step 4.1

Cancel the common factor of $x - 6$ .

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Step 4.1.1

Cancel the common factor.

$\frac{(x - 6) (x + 4)}{(x + 3) (x + 4)} \cdot \frac{(x + 1) (x - 1)}{x - 6}$

Step 4.1.2

Rewrite the expression.

$\frac{x + 4}{(x + 3) (x + 4)} \cdot ((x + 1) (x - 1))$

Step 4.2

Cancel the common factor of $x + 4$ .

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Step 4.2.1

Cancel the common factor.

$\frac{x + 4}{(x + 3) (x + 4)} \cdot ((x + 1) (x - 1))$

Step 4.2.2

Rewrite the expression.

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

Step 5

Expand $(x + 1) (x - 1)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x$ by $x$ .

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

Step 6.1.2

Move $- 1$ to the left of $x$ .

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

Step 6.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 6.1.4

Multiply $x$ by $1$ .

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

Step 6.1.5

Multiply $- 1$ by $1$ .

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

Step 6.2

Add $- x$ and $x$ .

$\frac{1}{x + 3} \cdot (x^{2} + 0 - 1)$

Step 6.3

Add $x^{2}$ and $0$ .

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

Step 7

Multiply $\frac{1}{x + 3}$ by $x^{2} - 1$ .

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

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 $b = 1$ .

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