Algebra Examples

$\frac{x^{2} + 14 x + 40}{9 x - x^{2}} \cdot \frac{3}{x^{2} + 6 x + 8} \cdot \frac{x^{2} - 7 x - 18}{- x - 10}$

Step 1

Factor $x^{2} - 7 x - 18$ using the AC method.

Tap for more steps...

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 $- 18$ and whose sum is $- 7$ .

$- 9, 2$

Step 1.2

Write the factored form using these integers.

$\frac{x^{2} + 14 x + 40}{9 x - x^{2}} \cdot \frac{3}{x^{2} + 6 x + 8} \cdot \frac{(x - 9) (x + 2)}{- x - 10}$

Step 2

Factor $x^{2} + 14 x + 40$ 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 $40$ and whose sum is $14$ .

$4, 10$

Step 2.2

Write the factored form using these integers.

$\frac{(x + 4) (x + 10)}{9 x - x^{2}} \cdot \frac{3}{x^{2} + 6 x + 8} \cdot \frac{(x - 9) (x + 2)}{- x - 10}$

Step 3

Factor $x$ out of $9 x - x^{2}$ .

Tap for more steps...

Step 3.1

Factor $x$ out of $9 x$ .

$\frac{(x + 4) (x + 10)}{x \cdot 9 - x^{2}} \cdot \frac{3}{x^{2} + 6 x + 8} \cdot \frac{(x - 9) (x + 2)}{- x - 10}$

Step 3.2

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

$\frac{(x + 4) (x + 10)}{x \cdot 9 + x (- x)} \cdot \frac{3}{x^{2} + 6 x + 8} \cdot \frac{(x - 9) (x + 2)}{- x - 10}$

Step 3.3

Factor $x$ out of $x \cdot 9 + x (- x)$ .

$\frac{(x + 4) (x + 10)}{x (9 - x)} \cdot \frac{3}{x^{2} + 6 x + 8} \cdot \frac{(x - 9) (x + 2)}{- x - 10}$

Step 4

Factor $x^{2} + 6 x + 8$ using the AC method.

Tap for more steps...

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 $8$ and whose sum is $6$ .

$2, 4$

Step 4.2

Write the factored form using these integers.

$\frac{(x + 4) (x + 10)}{x (9 - x)} \cdot \frac{3}{(x + 2) (x + 4)} \cdot \frac{(x - 9) (x + 2)}{- x - 10}$

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Cancel the common factor of $x + 4$ .

Tap for more steps...

Step 5.1.1

Factor $x + 4$ out of $(x + 2) (x + 4)$ .

$\frac{(x + 4) (x + 10)}{x (9 - x)} \cdot \frac{3}{(x + 4) (x + 2)} \cdot \frac{(x - 9) (x + 2)}{- x - 10}$

Step 5.1.2

Cancel the common factor.

$\frac{(x + 4) (x + 10)}{x (9 - x)} \cdot \frac{3}{(x + 4) (x + 2)} \cdot \frac{(x - 9) (x + 2)}{- x - 10}$

Step 5.1.3

Rewrite the expression.

$\frac{x + 10}{x (9 - x)} \cdot \frac{3}{x + 2} \cdot \frac{(x - 9) (x + 2)}{- x - 10}$

Step 5.2

Multiply $\frac{x + 10}{x (9 - x)}$ by $\frac{3}{x + 2}$ .

$\frac{(x + 10) \cdot 3}{x (9 - x) (x + 2)} \cdot \frac{(x - 9) (x + 2)}{- x - 10}$

Step 5.3

Cancel the common factor of $x + 2$ .

Tap for more steps...

Step 5.3.1

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

$\frac{(x + 10) \cdot 3}{(x + 2) (x (9 - x))} \cdot \frac{(x - 9) (x + 2)}{- x - 10}$

Step 5.3.2

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

$\frac{(x + 10) \cdot 3}{(x + 2) (x (9 - x))} \cdot \frac{(x + 2) (x - 9)}{- x - 10}$

Step 5.3.3

Cancel the common factor.

$\frac{(x + 10) \cdot 3}{(x + 2) (x (9 - x))} \cdot \frac{(x + 2) (x - 9)}{- x - 10}$

Step 5.3.4

Rewrite the expression.

$\frac{(x + 10) \cdot 3}{x (9 - x)} \cdot \frac{x - 9}{- x - 10}$

Step 5.4

Multiply $\frac{(x + 10) \cdot 3}{x (9 - x)}$ by $\frac{x - 9}{- x - 10}$ .

$\frac{(x + 10) \cdot 3 (x - 9)}{x (9 - x) (- x - 10)}$

Step 5.5

Cancel the common factor of $x + 10$ and $- x - 10$ .

Tap for more steps...

Step 5.5.1

Factor $- 1$ out of $x$ .

$\frac{(- 1 (- x) + 10) \cdot 3 (x - 9)}{x (9 - x) (- x - 10)}$

Step 5.5.2

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{(- 1 (- x) - 1 (- 10)) \cdot 3 (x - 9)}{x (9 - x) (- x - 10)}$

Step 5.5.3

Factor $- 1$ out of $- 1 (- x) - 1 (- 10)$ .

$\frac{- 1 (- x - 10) \cdot 3 (x - 9)}{x (9 - x) (- x - 10)}$

Step 5.5.4

Cancel the common factor.

$\frac{- 1 (- x - 10) \cdot 3 (x - 9)}{x (9 - x) (- x - 10)}$

Step 5.5.5

Rewrite the expression.

$\frac{- 1 \cdot 3 (x - 9)}{x (9 - x)}$

Step 5.6

Cancel the common factor of $x - 9$ and $9 - x$ .

Tap for more steps...

Step 5.6.1

Factor $- 1$ out of $x$ .

$\frac{- 1 \cdot 3 (- 1 (- x) - 9)}{x (9 - x)}$

Step 5.6.2

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

$\frac{- 1 \cdot 3 (- 1 (- x) - 1 (9))}{x (9 - x)}$

Step 5.6.3

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

$\frac{- 1 \cdot 3 (- 1 (- x + 9))}{x (9 - x)}$

Step 5.6.4

Reorder terms.

$\frac{- 1 \cdot 3 (- 1 (- x + 9))}{x (- x + 9)}$

Step 5.6.5

Cancel the common factor.

$\frac{- 1 \cdot 3 (- 1 (- x + 9))}{x (- x + 9)}$

Step 5.6.6

Rewrite the expression.

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

Step 6

Simplify the numerator.

Tap for more steps...

Step 6.1

Multiply $- 1$ by $3$ .

$\frac{- 3 \cdot - 1}{x}$

Step 6.2

Multiply $- 3$ by $- 1$ .

$\frac{3}{x}$