Algebra Examples

$\frac{x^{2} + 9 x - 22}{x^{2} - 121} \div (2 - x)$

Step 1

Rewrite the division as a fraction.

$\frac{\frac{x^{2} + 9 x - 22}{x^{2} - 121}}{2 - x}$

Step 2

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} + 9 x - 22}{x^{2} - 121} \cdot \frac{1}{2 - x}$

Step 3

Factor $x^{2} + 9 x - 22$ using the AC method.

Tap for more steps...

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 $- 22$ and whose sum is $9$ .

$- 2, 11$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 2) (x + 11)}{x^{2} - 121} \cdot \frac{1}{2 - x}$

Step 4

Simplify the denominator.

Tap for more steps...

Step 4.1

Rewrite $121$ as $11^{2}$ .

$\frac{(x - 2) (x + 11)}{x^{2} - 11^{2}} \cdot \frac{1}{2 - x}$

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 $a = x$ and $b = 11$ .

$\frac{(x - 2) (x + 11)}{(x + 11) (x - 11)} \cdot \frac{1}{2 - x}$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Cancel the common factor of $x + 11$ .

Tap for more steps...

Step 5.1.1

Cancel the common factor.

$\frac{(x - 2) (x + 11)}{(x + 11) (x - 11)} \cdot \frac{1}{2 - x}$

Step 5.1.2

Rewrite the expression.

$\frac{x - 2}{x - 11} \cdot \frac{1}{2 - x}$

Step 5.2

Multiply $\frac{x - 2}{x - 11}$ by $\frac{1}{2 - x}$ .

$\frac{x - 2}{(x - 11) (2 - x)}$

Step 5.3

Cancel the common factor of $x - 2$ and $2 - x$ .

Tap for more steps...

Step 5.3.1

Factor $- 1$ out of $x$ .

$\frac{- 1 (- x) - 2}{(x - 11) (2 - x)}$

Step 5.3.2

Rewrite $- 2$ as $- 1 (2)$ .

$\frac{- 1 (- x) - 1 (2)}{(x - 11) (2 - x)}$

Step 5.3.3

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

$\frac{- 1 (- x + 2)}{(x - 11) (2 - x)}$

Step 5.3.4

Reorder terms.

$\frac{- 1 (- x + 2)}{(x - 11) (- x + 2)}$

Step 5.3.5

Cancel the common factor.

$\frac{- 1 (- x + 2)}{(x - 11) (- x + 2)}$

Step 5.3.6

Rewrite the expression.

$\frac{- 1}{x - 11}$

Step 5.4

Move the negative in front of the fraction.

$- \frac{1}{x - 11}$