Algebra Examples

$\frac{- 2}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 16}{2 - x} \cdot \frac{18 - 3 x}{x^{2} + 4 x}$

Step 1

Simplify with factoring out.

Tap for more steps...

Step 1.1

Factor $3$ out of $18 - 3 x$ .

Tap for more steps...

Step 1.1.1

Factor $3$ out of $18$ .

$\frac{- 2}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 16}{2 - x} \cdot \frac{3 (6) - 3 x}{x^{2} + 4 x}$

Step 1.1.2

Factor $3$ out of $- 3 x$ .

$\frac{- 2}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 16}{2 - x} \cdot \frac{3 (6) + 3 (- x)}{x^{2} + 4 x}$

Step 1.1.3

Factor $3$ out of $3 (6) + 3 (- x)$ .

$\frac{- 2}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 16}{2 - x} \cdot \frac{3 (6 - x)}{x^{2} + 4 x}$

Step 1.2

Factor $x$ out of $x^{2} + 4 x$ .

Tap for more steps...

Step 1.2.1

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

$\frac{- 2}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 16}{2 - x} \cdot \frac{3 (6 - x)}{x \cdot x + 4 x}$

Step 1.2.2

Factor $x$ out of $4 x$ .

$\frac{- 2}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 16}{2 - x} \cdot \frac{3 (6 - x)}{x \cdot x + x \cdot 4}$

Step 1.2.3

Factor $x$ out of $x \cdot x + x \cdot 4$ .

$\frac{- 2}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 16}{2 - x} \cdot \frac{3 (6 - x)}{x (x + 4)}$

Step 2

Factor $x^{2} - 10 x + 24$ 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 $24$ and whose sum is $- 10$ .

$- 6, - 4$

Step 2.2

Write the factored form using these integers.

$\frac{- 2}{(x - 6) (x - 4)} \cdot \frac{x^{2} - 16}{2 - x} \cdot \frac{3 (6 - x)}{x (x + 4)}$

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

Rewrite $16$ as $4^{2}$ .

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

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 $b = 4$ .

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

Step 4

Simplify terms.

Tap for more steps...

Step 4.1

Cancel the common factor of $x - 4$ .

Tap for more steps...

Step 4.1.1

Factor $x - 4$ out of $(x - 6) (x - 4)$ .

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

Step 4.1.2

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

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

Step 4.1.3

Cancel the common factor.

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

Step 4.1.4

Rewrite the expression.

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

Step 4.2

Multiply $\frac{- 2}{x - 6}$ by $\frac{x + 4}{2 - x}$ .

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

Step 4.3

Cancel the common factor of $x + 4$ .

Tap for more steps...

Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Cancel the common factor.

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

Step 4.3.4

Rewrite the expression.

$\frac{- 2}{(x - 6) (2 - x)} \cdot \frac{3 (6 - x)}{x}$

Step 4.4

Multiply $\frac{- 2}{(x - 6) (2 - x)}$ by $\frac{3 (6 - x)}{x}$ .

$\frac{- 2 (3 (6 - x))}{(x - 6) (2 - x) x}$

Step 4.5

Multiply $3$ by $- 2$ .

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

Step 4.6

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

Tap for more steps...

Step 4.6.1

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

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

Step 4.6.2

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

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

Step 4.6.3

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

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

Step 4.6.4

Reorder terms.

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

Step 4.6.5

Cancel the common factor.

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

Step 4.6.6

Rewrite the expression.

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

Step 5

Multiply $- 6$ by $- 1$ .

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

Step 6

Reorder factors in $\frac{6}{(2 - x) x}$ .

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