Algebra Examples

$\frac{x^{2} + 11 x + 18}{x^{2} + x} \cdot \frac{1 - x^{2}}{x^{2} + 8 x - 9} \div \frac{x^{2} - 4 x - 12}{36 - x^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} + 11 x + 18}{x^{2} + x} \cdot \frac{1 - x^{2}}{x^{2} + 8 x - 9} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 2

Factor $x^{2} + 11 x + 18$ using the AC method.

Tap for more steps...

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 $18$ and whose sum is $11$ .

$2, 9$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 9)}{x^{2} + x} \cdot \frac{1 - x^{2}}{x^{2} + 8 x - 9} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 3

Factor $x$ out of $x^{2} + x$ .

Tap for more steps...

Step 3.1

Factor $x$ out of $x^{2}$ .

$\frac{(x + 2) (x + 9)}{x \cdot x + x} \cdot \frac{1 - x^{2}}{x^{2} + 8 x - 9} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 3.2

Raise $x$ to the power of $1$ .

$\frac{(x + 2) (x + 9)}{x \cdot x + x^{1}} \cdot \frac{1 - x^{2}}{x^{2} + 8 x - 9} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 3.3

Factor $x$ out of $x^{1}$ .

$\frac{(x + 2) (x + 9)}{x \cdot x + x \cdot 1} \cdot \frac{1 - x^{2}}{x^{2} + 8 x - 9} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 3.4

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$\frac{(x + 2) (x + 9)}{x (x + 1)} \cdot \frac{1 - x^{2}}{x^{2} + 8 x - 9} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

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

$\frac{(x + 2) (x + 9)}{x (x + 1)} \cdot \frac{1^{2} - x^{2}}{x^{2} + 8 x - 9} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$\frac{(x + 2) (x + 9)}{x (x + 1)} \cdot \frac{(1 + x) (1 - x)}{x^{2} + 8 x - 9} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 5

Factor $x^{2} + 8 x - 9$ using the AC method.

Tap for more steps...

Step 5.1

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 $- 9$ and whose sum is $8$ .

$- 1, 9$

Step 5.2

Write the factored form using these integers.

$\frac{(x + 2) (x + 9)}{x (x + 1)} \cdot \frac{(1 + x) (1 - x)}{(x - 1) (x + 9)} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6

Simplify terms.

Tap for more steps...

Step 6.1

Cancel the common factor of $x + 9$ .

Tap for more steps...

Step 6.1.1

Factor $x + 9$ out of $(x + 2) (x + 9)$ .

$\frac{(x + 9) (x + 2)}{x (x + 1)} \cdot \frac{(1 + x) (1 - x)}{(x - 1) (x + 9)} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.1.2

Factor $x + 9$ out of $(x - 1) (x + 9)$ .

$\frac{(x + 9) (x + 2)}{x (x + 1)} \cdot \frac{(1 + x) (1 - x)}{(x + 9) (x - 1)} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.1.3

Cancel the common factor.

$\frac{(x + 9) (x + 2)}{x (x + 1)} \cdot \frac{(1 + x) (1 - x)}{(x + 9) (x - 1)} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.1.4

Rewrite the expression.

$\frac{x + 2}{x (x + 1)} \cdot \frac{(1 + x) (1 - x)}{x - 1} \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.2

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

$\frac{(x + 2) ((1 + x) (1 - x))}{x (x + 1) (x - 1)} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.3

Cancel the common factor of $1 + x$ and $x + 1$ .

Tap for more steps...

Step 6.3.1

Reorder terms.

$\frac{(x + 2) ((x + 1) (1 - x))}{x (x + 1) (x - 1)} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.3.2

Cancel the common factor.

$\frac{(x + 2) ((x + 1) (1 - x))}{x (x + 1) (x - 1)} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.3.3

Rewrite the expression.

$\frac{(x + 2) (1 - x)}{x (x - 1)} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.4

Cancel the common factor of $1 - x$ and $x - 1$ .

Tap for more steps...

Step 6.4.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{(x + 2) (- 1 (- 1) - x)}{x (x - 1)} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.4.2

Factor $- 1$ out of $- x$ .

$\frac{(x + 2) (- 1 (- 1) - (x))}{x (x - 1)} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.4.3

Factor $- 1$ out of $- 1 (- 1) - (x)$ .

$\frac{(x + 2) (- 1 (- 1 + x))}{x (x - 1)} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.4.4

Reorder terms.

$\frac{(x + 2) (- 1 (x - 1))}{x (x - 1)} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.4.5

Cancel the common factor.

$\frac{(x + 2) (- 1 (x - 1))}{x (x - 1)} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.4.6

Rewrite the expression.

$\frac{(x + 2) \cdot (- 1)}{x} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.5

Simplify the expression.

Tap for more steps...

Step 6.5.1

Move $- 1$ to the left of $x + 2$ .

$\frac{- 1 \cdot (x + 2)}{x} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 6.5.2

Move the negative in front of the fraction.

$- \frac{x + 2}{x} \cdot \frac{36 - x^{2}}{x^{2} - 4 x - 12}$

Step 7

Simplify the numerator.

Tap for more steps...

Step 7.1

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

$- \frac{x + 2}{x} \cdot \frac{6^{2} - x^{2}}{x^{2} - 4 x - 12}$

Step 7.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 6$ and $b = x$ .

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

Step 8

Factor $x^{2} - 4 x - 12$ using the AC method.

Tap for more steps...

Step 8.1

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 $- 4$ .

$- 6, 2$

Step 8.2

Write the factored form using these integers.

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

Step 9

Simplify terms.

Tap for more steps...

Step 9.1

Cancel the common factor of $x + 2$ .

Tap for more steps...

Step 9.1.1

Move the leading negative in $- \frac{x + 2}{x}$ into the numerator.

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

Step 9.1.2

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

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

Step 9.1.3

Factor $x + 2$ out of $(x - 6) (x + 2)$ .

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

Step 9.1.4

Cancel the common factor.

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

Step 9.1.5

Rewrite the expression.

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

Step 9.2

Multiply $\frac{- 1}{x}$ by $\frac{(6 + x) (6 - x)}{x - 6}$ .

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

Step 9.3

Cancel the common factor of $6 - x$ and $x - 6$ .

Tap for more steps...

Step 9.3.1

Rewrite $6$ as $- 1 (- 6)$ .

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

Step 9.3.2

Factor $- 1$ out of $- x$ .

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

Step 9.3.3

Factor $- 1$ out of $- 1 (- 6) - (x)$ .

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

Step 9.3.4

Reorder terms.

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

Step 9.3.5

Cancel the common factor.

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

Step 9.3.6

Rewrite the expression.

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

Step 10

Simplify the numerator.

Tap for more steps...

Step 10.1

Multiply $- 1$ by $- 1$ .

$\frac{1 (6 + x)}{x}$

Step 10.2

Multiply $6 + x$ by $1$ .

$\frac{6 + x}{x}$