Algebra Examples

$\frac{- x - 6}{x + 9} \cdot \frac{x + 10}{- 2 x - 18} \div \frac{x^{2} + 16 x + 60}{x^{2} + 18 x + 81}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{- x - 6}{x + 9} \cdot \frac{x + 10}{- 2 x - 18} \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 2

Factor $2$ out of $- 2 x - 18$ .

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Step 2.1

Factor $2$ out of $- 2 x$ .

$\frac{- x - 6}{x + 9} \cdot \frac{x + 10}{2 (- x) - 18} \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 2.2

Factor $2$ out of $- 18$ .

$\frac{- x - 6}{x + 9} \cdot \frac{x + 10}{2 (- x) + 2 (- 9)} \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 2.3

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

$\frac{- x - 6}{x + 9} \cdot \frac{x + 10}{2 (- x - 9)} \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 3

Multiply $\frac{- x - 6}{x + 9} \cdot \frac{x + 10}{2 (- x - 9)}$ .

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Step 3.1

Multiply $\frac{- x - 6}{x + 9}$ by $\frac{x + 10}{2 (- x - 9)}$ .

$\frac{(- x - 6) (x + 10)}{(x + 9) (2 (- x - 9))} \cdot \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 3.2

Factor $- 1$ out of $x$ .

$\frac{(- x - 6) (x + 10)}{(- 1 (- x) + 9) (2 (- x - 9))} \cdot \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 3.3

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{(- x - 6) (x + 10)}{(- 1 (- x) - 1 (- 9)) (2 (- x - 9))} \cdot \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 3.4

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

$\frac{(- x - 6) (x + 10)}{- 1 (- x - 9) (2 (- x - 9))} \cdot \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 3.5

Raise $- x - 9$ to the power of $1$ .

$\frac{(- x - 6) (x + 10)}{- 1 \cdot 2 ({(- x - 9)}^{1} (- x - 9))} \cdot \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 3.6

Raise $- x - 9$ to the power of $1$ .

$\frac{(- x - 6) (x + 10)}{- 1 \cdot 2 ({(- x - 9)}^{1} {(- x - 9)}^{1})} \cdot \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 3.7

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{(- x - 6) (x + 10)}{- 1 \cdot 2 {(- x - 9)}^{1 + 1}} \cdot \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 3.8

Add $1$ and $1$ .

$\frac{(- x - 6) (x + 10)}{- 1 \cdot 2 {(- x - 9)}^{2}} \cdot \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 3.9

Multiply $- 1$ by $2$ .

$\frac{(- x - 6) (x + 10)}{- 2 {(- x - 9)}^{2}} \cdot \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 4

Move the negative in front of the fraction.

$- \frac{(- x - 6) (x + 10)}{2 {(- x - 9)}^{2}} \cdot \frac{x^{2} + 18 x + 81}{x^{2} + 16 x + 60}$

Step 5

Factor using the perfect square rule.

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Step 5.1

Rewrite $81$ as $9^{2}$ .

$- \frac{(- x - 6) (x + 10)}{2 {(- x - 9)}^{2}} \cdot \frac{x^{2} + 18 x + 9^{2}}{x^{2} + 16 x + 60}$

Step 5.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$18 x = 2 \cdot x \cdot 9$

Step 5.3

Rewrite the polynomial.

$- \frac{(- x - 6) (x + 10)}{2 {(- x - 9)}^{2}} \cdot \frac{x^{2} + 2 \cdot x \cdot 9 + 9^{2}}{x^{2} + 16 x + 60}$

Step 5.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 9$ .

$- \frac{(- x - 6) (x + 10)}{2 {(- x - 9)}^{2}} \cdot \frac{{(x + 9)}^{2}}{x^{2} + 16 x + 60}$

Step 6

Factor $x^{2} + 16 x + 60$ using the AC method.

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Step 6.1

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 $60$ and whose sum is $16$ .

$6, 10$

Step 6.2

Write the factored form using these integers.

$- \frac{(- x - 6) (x + 10)}{2 {(- x - 9)}^{2}} \cdot \frac{{(x + 9)}^{2}}{(x + 6) (x + 10)}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $x + 10$ .

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Step 7.1.1

Move the leading negative in $- \frac{(- x - 6) (x + 10)}{2 {(- x - 9)}^{2}}$ into the numerator.

$\frac{- (- x - 6) (x + 10)}{2 {(- x - 9)}^{2}} \cdot \frac{{(x + 9)}^{2}}{(x + 6) (x + 10)}$

Step 7.1.2

Factor $x + 10$ out of $- (- x - 6) (x + 10)$ .

$\frac{(x + 10) (- (- x - 6))}{2 {(- x - 9)}^{2}} \cdot \frac{{(x + 9)}^{2}}{(x + 6) (x + 10)}$

Step 7.1.3

Factor $x + 10$ out of $(x + 6) (x + 10)$ .

$\frac{(x + 10) (- (- x - 6))}{2 {(- x - 9)}^{2}} \cdot \frac{{(x + 9)}^{2}}{(x + 10) (x + 6)}$

Step 7.1.4

Cancel the common factor.

$\frac{(x + 10) (- (- x - 6))}{2 {(- x - 9)}^{2}} \cdot \frac{{(x + 9)}^{2}}{(x + 10) (x + 6)}$

Step 7.1.5

Rewrite the expression.

$\frac{- (- x - 6)}{2 {(- x - 9)}^{2}} \cdot \frac{{(x + 9)}^{2}}{x + 6}$

Step 7.2

Multiply $\frac{- (- x - 6)}{2 {(- x - 9)}^{2}}$ by $\frac{{(x + 9)}^{2}}{x + 6}$ .

$\frac{- (- x - 6) {(x + 9)}^{2}}{2 {(- x - 9)}^{2} (x + 6)}$

Step 7.3

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

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Step 7.3.1

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

$\frac{- (- (x) - 6) {(x + 9)}^{2}}{2 {(- x - 9)}^{2} (x + 6)}$

Step 7.3.2

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

$\frac{- (- (x) - 1 (6)) {(x + 9)}^{2}}{2 {(- x - 9)}^{2} (x + 6)}$

Step 7.3.3

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

$\frac{- - (x + 6) {(x + 9)}^{2}}{2 {(- x - 9)}^{2} (x + 6)}$

Step 7.3.4

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

$\frac{- (- 1 (x + 6)) {(x + 9)}^{2}}{2 {(- x - 9)}^{2} (x + 6)}$

Step 7.3.5

Cancel the common factor.

$\frac{- (- 1 (x + 6)) {(x + 9)}^{2}}{2 {(- x - 9)}^{2} (x + 6)}$

Step 7.3.6

Rewrite the expression.

$\frac{- (- 1) {(x + 9)}^{2}}{2 {(- x - 9)}^{2}}$

Step 7.4

Cancel the common factor of ${(x + 9)}^{2}$ and ${(- x - 9)}^{2}$ .

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Step 7.4.1

Factor $- 1$ out of $x$ .

$\frac{- (- 1) {(- 1 (- x) + 9)}^{2}}{2 {(- x - 9)}^{2}}$

Step 7.4.2

Rewrite $9$ as $- 1 (- 9)$ .

$\frac{- (- 1) {(- 1 (- x) - 1 (- 9))}^{2}}{2 {(- x - 9)}^{2}}$

Step 7.4.3

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

$\frac{- (- 1) {(- 1 (- x - 9))}^{2}}{2 {(- x - 9)}^{2}}$

Step 7.4.4

Apply the product rule to $- 1 (- x - 9)$ .

$\frac{- (- 1) ({(- 1)}^{2} {(- x - 9)}^{2})}{2 {(- x - 9)}^{2}}$

Step 7.4.5

Raise $- 1$ to the power of $2$ .

$\frac{- (- 1) (1 {(- x - 9)}^{2})}{2 {(- x - 9)}^{2}}$

Step 7.4.6

Multiply ${(- x - 9)}^{2}$ by $1$ .

$\frac{- (- 1) {(- x - 9)}^{2}}{2 {(- x - 9)}^{2}}$

Step 7.4.7

Cancel the common factor.

$\frac{- (- 1) {(- x - 9)}^{2}}{2 {(- x - 9)}^{2}}$

Step 7.4.8

Rewrite the expression.

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

Step 7.5

Multiply $- 1$ by $- 1$ .

$\frac{1}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{1}{2}$

Decimal Form:

$0.5$