Algebra Examples

$\frac{4 x - 4 x^{2}}{x^{2} - 9 x + 8} \cdot \frac{x^{2} - 5 x - 24}{- 4 x}$

Step 1

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

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Step 1.1

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

$\frac{4 x (1) - 4 x^{2}}{x^{2} - 9 x + 8} \cdot \frac{x^{2} - 5 x - 24}{- 4 x}$

Step 1.2

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

$\frac{4 x (1) + 4 x (- x)}{x^{2} - 9 x + 8} \cdot \frac{x^{2} - 5 x - 24}{- 4 x}$

Step 1.3

Factor $4 x$ out of $4 x (1) + 4 x (- x)$ .

$\frac{4 x (1 - x)}{x^{2} - 9 x + 8} \cdot \frac{x^{2} - 5 x - 24}{- 4 x}$

Step 2

Factor $x^{2} - 9 x + 8$ using the AC method.

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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 $8$ and whose sum is $- 9$ .

$- 8, - 1$

Step 2.2

Write the factored form using these integers.

$\frac{4 x (1 - x)}{(x - 8) (x - 1)} \cdot \frac{x^{2} - 5 x - 24}{- 4 x}$

Step 3

Factor $x^{2} - 5 x - 24$ 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 $- 24$ and whose sum is $- 5$ .

$- 8, 3$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $4 x$ .

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Step 4.1.1

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

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

Step 4.1.2

Cancel the common factor.

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

Step 4.1.3

Rewrite the expression.

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

Step 4.2

Cancel the common factor of $x - 8$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

Multiply $\frac{1 - x}{x - 1}$ by $\frac{x + 3}{- 1}$ .

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

Step 4.4

Cancel the common factor of $1 - x$ and $x - 1$ .

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Step 4.4.1

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

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

Step 4.4.2

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

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

Step 4.4.3

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

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

Step 4.4.4

Reorder terms.

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

Step 4.4.5

Cancel the common factor.

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

Step 4.4.6

Rewrite the expression.

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

Step 4.4.7

Move the negative one from the denominator of $\frac{- 1 (x + 3)}{- 1}$ .

$- 1 \cdot (- 1 (x + 3))$

Step 4.5

Rewrite $- 1 \cdot (- 1 (x + 3))$ as $- (- 1 (x + 3))$ .

$- (- 1 (x + 3))$

Step 4.6

Apply the distributive property.

$- (- 1 x - 1 \cdot 3)$

Step 4.7

Simplify the expression.

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Step 4.7.1

Rewrite $- 1 x$ as $- x$ .

$- (- x - 1 \cdot 3)$

Step 4.7.2

Multiply $- 1$ by $3$ .

$- (- x - 3)$

Step 4.8

Apply the distributive property.

$- - x - - 3$

Step 5

Multiply $- - x$ .

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Step 5.1

Multiply $- 1$ by $- 1$ .

$1 x - - 3$

Step 5.2

Multiply $x$ by $1$ .

$x - - 3$

Step 6

Multiply $- 1$ by $- 3$ .

$x + 3$