Algebra Examples

$\frac{- 4 x - 32}{x^{2} - 12 x + 36} \cdot \frac{x^{2} - 9 x + 18}{3 x + 24}$

Step 1

Factor $4$ out of $- 4 x - 32$ .

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Step 1.1

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

$\frac{4 (- x) - 32}{x^{2} - 12 x + 36} \cdot \frac{x^{2} - 9 x + 18}{3 x + 24}$

Step 1.2

Factor $4$ out of $- 32$ .

$\frac{4 (- x) + 4 (- 8)}{x^{2} - 12 x + 36} \cdot \frac{x^{2} - 9 x + 18}{3 x + 24}$

Step 1.3

Factor $4$ out of $4 (- x) + 4 (- 8)$ .

$\frac{4 (- x - 8)}{x^{2} - 12 x + 36} \cdot \frac{x^{2} - 9 x + 18}{3 x + 24}$

Step 2

Factor using the perfect square rule.

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Step 2.1

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

$\frac{4 (- x - 8)}{x^{2} - 12 x + 6^{2}} \cdot \frac{x^{2} - 9 x + 18}{3 x + 24}$

Step 2.2

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

$12 x = 2 \cdot x \cdot 6$

Step 2.3

Rewrite the polynomial.

$\frac{4 (- x - 8)}{x^{2} - 2 \cdot x \cdot 6 + 6^{2}} \cdot \frac{x^{2} - 9 x + 18}{3 x + 24}$

Step 2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 6$ .

$\frac{4 (- x - 8)}{{(x - 6)}^{2}} \cdot \frac{x^{2} - 9 x + 18}{3 x + 24}$

Step 3

Factor $x^{2} - 9 x + 18$ using the AC method.

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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 $18$ and whose sum is $- 9$ .

$- 6, - 3$

Step 3.2

Write the factored form using these integers.

$\frac{4 (- x - 8)}{{(x - 6)}^{2}} \cdot \frac{(x - 6) (x - 3)}{3 x + 24}$

Step 4

Simplify terms.

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Step 4.1

Factor $3$ out of $3 x + 24$ .

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Step 4.1.1

Factor $3$ out of $3 x$ .

$\frac{4 (- x - 8)}{{(x - 6)}^{2}} \cdot \frac{(x - 6) (x - 3)}{3 (x) + 24}$

Step 4.1.2

Factor $3$ out of $24$ .

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

Step 4.1.3

Factor $3$ out of $3 x + 3 \cdot 8$ .

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

Step 4.2

Combine.

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

Step 4.3

Cancel the common factor of $x - 6$ and ${(x - 6)}^{2}$ .

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Step 4.3.1

Factor $x - 6$ out of $4 (- x - 8) ((x - 6) (x - 3))$ .

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $x - 6$ out of ${(x - 6)}^{2} (3 (x + 8))$ .

$\frac{(x - 6) (4 (- x - 8) (x - 3))}{(x - 6) ((x - 6) (3 (x + 8)))}$

Step 4.3.2.2

Cancel the common factor.

$\frac{(x - 6) (4 (- x - 8) (x - 3))}{(x - 6) ((x - 6) (3 (x + 8)))}$

Step 4.3.2.3

Rewrite the expression.

$\frac{4 (- x - 8) (x - 3)}{(x - 6) (3 (x + 8))}$

Step 4.4

Cancel the common factor of $- x - 8$ and $x + 8$ .

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Step 4.4.1

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

$\frac{4 (- (x) - 8) (x - 3)}{(x - 6) (3 (x + 8))}$

Step 4.4.2

Rewrite $- 8$ as $- 1 (8)$ .

$\frac{4 (- (x) - 1 (8)) (x - 3)}{(x - 6) (3 (x + 8))}$

Step 4.4.3

Factor $- 1$ out of $- (x) - 1 (8)$ .

$\frac{4 (- (x + 8)) (x - 3)}{(x - 6) (3 (x + 8))}$

Step 4.4.4

Rewrite $- (x + 8)$ as $- 1 (x + 8)$ .

$\frac{4 (- 1 (x + 8)) (x - 3)}{(x - 6) (3 (x + 8))}$

Step 4.4.5

Cancel the common factor.

$\frac{4 (- 1 (x + 8)) (x - 3)}{(x - 6) \cdot 3 (x + 8)}$

Step 4.4.6

Rewrite the expression.

$\frac{4 \cdot (- 1) (x - 3)}{(x - 6) \cdot 3}$

Step 4.5

Simplify the expression.

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Step 4.5.1

Multiply $4$ by $- 1$ .

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

Step 4.5.2

Move $3$ to the left of $x - 6$ .

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

Step 4.5.3

Move the negative in front of the fraction.

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