Algebra Examples

$\frac{x^{4} - 6 x^{3}}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 6 x - 36}{x^{2} - 36}$

Step 1

Factor $x^{3}$ out of $x^{4} - 6 x^{3}$ .

Tap for more steps...

Step 1.1

Factor $x^{3}$ out of $x^{4}$ .

$\frac{x^{3} x - 6 x^{3}}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 6 x - 36}{x^{2} - 36}$

Step 1.2

Factor $x^{3}$ out of $- 6 x^{3}$ .

$\frac{x^{3} x + x^{3} \cdot - 6}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 6 x - 36}{x^{2} - 36}$

Step 1.3

Factor $x^{3}$ out of $x^{3} x + x^{3} \cdot - 6$ .

$\frac{x^{3} (x - 6)}{4 x^{2} - 20 x + 24} \cdot \frac{2 x^{2} + 6 x - 36}{x^{2} - 36}$

Step 2

Simplify the denominator.

Tap for more steps...

Step 2.1

Factor $4$ out of $4 x^{2} - 20 x + 24$ .

Tap for more steps...

Step 2.1.1

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

$\frac{x^{3} (x - 6)}{4 (x^{2}) - 20 x + 24} \cdot \frac{2 x^{2} + 6 x - 36}{x^{2} - 36}$

Step 2.1.2

Factor $4$ out of $- 20 x$ .

$\frac{x^{3} (x - 6)}{4 (x^{2}) + 4 (- 5 x) + 24} \cdot \frac{2 x^{2} + 6 x - 36}{x^{2} - 36}$

Step 2.1.3

Factor $4$ out of $24$ .

$\frac{x^{3} (x - 6)}{4 x^{2} + 4 (- 5 x) + 4 \cdot 6} \cdot \frac{2 x^{2} + 6 x - 36}{x^{2} - 36}$

Step 2.1.4

Factor $4$ out of $4 x^{2} + 4 (- 5 x)$ .

$\frac{x^{3} (x - 6)}{4 (x^{2} - 5 x) + 4 \cdot 6} \cdot \frac{2 x^{2} + 6 x - 36}{x^{2} - 36}$

Step 2.1.5

Factor $4$ out of $4 (x^{2} - 5 x) + 4 \cdot 6$ .

$\frac{x^{3} (x - 6)}{4 (x^{2} - 5 x + 6)} \cdot \frac{2 x^{2} + 6 x - 36}{x^{2} - 36}$

Step 2.2

Factor $x^{2} - 5 x + 6$ using the AC method.

Tap for more steps...

Step 2.2.1

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 $6$ and whose sum is $- 5$ .

$- 3, - 2$

Step 2.2.2

Write the factored form using these integers.

$\frac{x^{3} (x - 6)}{4 ((x - 3) (x - 2))} \cdot \frac{2 x^{2} + 6 x - 36}{x^{2} - 36}$

$\frac{x^{3} (x - 6)}{4 (x - 3) (x - 2)} \cdot \frac{2 x^{2} + 6 x - 36}{x^{2} - 36}$

Step 3

Simplify the numerator.

Tap for more steps...

Step 3.1

Factor $2$ out of $2 x^{2} + 6 x - 36$ .

Tap for more steps...

Step 3.1.1

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

$\frac{x^{3} (x - 6)}{4 (x - 3) (x - 2)} \cdot \frac{2 (x^{2}) + 6 x - 36}{x^{2} - 36}$

Step 3.1.2

Factor $2$ out of $6 x$ .

$\frac{x^{3} (x - 6)}{4 (x - 3) (x - 2)} \cdot \frac{2 (x^{2}) + 2 (3 x) - 36}{x^{2} - 36}$

Step 3.1.3

Factor $2$ out of $- 36$ .

$\frac{x^{3} (x - 6)}{4 (x - 3) (x - 2)} \cdot \frac{2 x^{2} + 2 (3 x) + 2 \cdot - 18}{x^{2} - 36}$

Step 3.1.4

Factor $2$ out of $2 x^{2} + 2 (3 x)$ .

$\frac{x^{3} (x - 6)}{4 (x - 3) (x - 2)} \cdot \frac{2 (x^{2} + 3 x) + 2 \cdot - 18}{x^{2} - 36}$

Step 3.1.5

Factor $2$ out of $2 (x^{2} + 3 x) + 2 \cdot - 18$ .

$\frac{x^{3} (x - 6)}{4 (x - 3) (x - 2)} \cdot \frac{2 (x^{2} + 3 x - 18)}{x^{2} - 36}$

Step 3.2

Factor $x^{2} + 3 x - 18$ using the AC method.

Tap for more steps...

Step 3.2.1

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 $3$ .

$- 3, 6$

Step 3.2.2

Write the factored form using these integers.

$\frac{x^{3} (x - 6)}{4 (x - 3) (x - 2)} \cdot \frac{2 ((x - 3) (x + 6))}{x^{2} - 36}$

$\frac{x^{3} (x - 6)}{4 (x - 3) (x - 2)} \cdot \frac{2 (x - 3) (x + 6)}{x^{2} - 36}$

Step 4

Simplify the denominator.

Tap for more steps...

Step 4.1

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

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

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 $b = 6$ .

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

Step 5

Simplify terms.

Tap for more steps...

Step 5.1

Combine.

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

Step 5.2

Cancel the common factor of $x - 6$ .

Tap for more steps...

Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Rewrite the expression.

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

Step 5.3

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 5.3.1

Factor $2$ out of $x^{3} (2 (x - 3) (x + 6))$ .

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

Step 5.3.2

Cancel the common factors.

Tap for more steps...

Step 5.3.2.1

Factor $2$ out of $4 (x - 3) (x - 2) (x + 6)$ .

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

Step 5.3.2.2

Cancel the common factor.

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

Step 5.3.2.3

Rewrite the expression.

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

Step 5.4

Cancel the common factor of $x - 3$ .

Tap for more steps...

Step 5.4.1

Cancel the common factor.

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

Step 5.4.2

Rewrite the expression.

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

Step 5.5

Cancel the common factor of $x + 6$ .

Tap for more steps...

Step 5.5.1

Cancel the common factor.

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

Step 5.5.2

Rewrite the expression.

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