Algebra Examples

$\frac{24 - 6 x}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 8 x + 12}{10 - x} \div \frac{x^{2} + 8 x - 20}{100 x - x^{3}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{24 - 6 x}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 8 x + 12}{10 - x} \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 2

Factor $6$ out of $24 - 6 x$ .

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Step 2.1

Factor $6$ out of $24$ .

$\frac{6 (4) - 6 x}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 8 x + 12}{10 - x} \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 2.2

Factor $6$ out of $- 6 x$ .

$\frac{6 (4) + 6 (- x)}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 8 x + 12}{10 - x} \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 2.3

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

$\frac{6 (4 - x)}{x^{2} - 10 x + 24} \cdot \frac{x^{2} - 8 x + 12}{10 - x} \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 3

Factor $x^{2} - 10 x + 24$ 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 $- 10$ .

$- 6, - 4$

Step 3.2

Write the factored form using these integers.

$\frac{6 (4 - x)}{(x - 6) (x - 4)} \cdot \frac{x^{2} - 8 x + 12}{10 - x} \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 4

Factor $x^{2} - 8 x + 12$ using the AC method.

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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 $12$ and whose sum is $- 8$ .

$- 6, - 2$

Step 4.2

Write the factored form using these integers.

$\frac{6 (4 - x)}{(x - 6) (x - 4)} \cdot \frac{(x - 6) (x - 2)}{10 - x} \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x - 6$ .

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Step 5.1.1

Cancel the common factor.

$\frac{6 (4 - x)}{(x - 6) (x - 4)} \cdot \frac{(x - 6) (x - 2)}{10 - x} \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 5.1.2

Rewrite the expression.

$\frac{6 (4 - x)}{x - 4} \cdot \frac{x - 2}{10 - x} \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 5.2

Multiply $\frac{6 (4 - x)}{x - 4}$ by $\frac{x - 2}{10 - x}$ .

$\frac{6 (4 - x) (x - 2)}{(x - 4) (10 - x)} \cdot \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 5.3

Cancel the common factor of $4 - x$ and $x - 4$ .

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Step 5.3.1

Rewrite $4$ as $- 1 (- 4)$ .

$\frac{6 (- 1 (- 4) - x) (x - 2)}{(x - 4) (10 - x)} \cdot \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 5.3.2

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

$\frac{6 (- 1 (- 4) - (x)) (x - 2)}{(x - 4) (10 - x)} \cdot \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 5.3.3

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

$\frac{6 (- 1 (- 4 + x)) (x - 2)}{(x - 4) (10 - x)} \cdot \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 5.3.4

Reorder terms.

$\frac{6 (- 1 (x - 4)) (x - 2)}{(x - 4) (10 - x)} \cdot \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 5.3.5

Cancel the common factor.

$\frac{6 (- 1 (x - 4)) (x - 2)}{(x - 4) (10 - x)} \cdot \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 5.3.6

Rewrite the expression.

$\frac{6 \cdot (- 1) (x - 2)}{10 - x} \cdot \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 5.4

Simplify the expression.

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Step 5.4.1

Multiply $6$ by $- 1$ .

$\frac{- 6 (x - 2)}{10 - x} \cdot \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 5.4.2

Move the negative in front of the fraction.

$- \frac{6 (x - 2)}{10 - x} \cdot \frac{100 x - x^{3}}{x^{2} + 8 x - 20}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $x$ out of $100 x - x^{3}$ .

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Step 6.1.1

Factor $x$ out of $100 x$ .

$- \frac{6 (x - 2)}{10 - x} \cdot \frac{x \cdot 100 - x^{3}}{x^{2} + 8 x - 20}$

Step 6.1.2

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

$- \frac{6 (x - 2)}{10 - x} \cdot \frac{x \cdot 100 + x (- x^{2})}{x^{2} + 8 x - 20}$

Step 6.1.3

Factor $x$ out of $x \cdot 100 + x (- x^{2})$ .

$- \frac{6 (x - 2)}{10 - x} \cdot \frac{x (100 - x^{2})}{x^{2} + 8 x - 20}$

Step 6.2

Rewrite $100$ as $10^{2}$ .

$- \frac{6 (x - 2)}{10 - x} \cdot \frac{x (10^{2} - x^{2})}{x^{2} + 8 x - 20}$

Step 6.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 10$ and $b = x$ .

$- \frac{6 (x - 2)}{10 - x} \cdot \frac{x (10 + x) (10 - x)}{x^{2} + 8 x - 20}$

Step 7

Factor $x^{2} + 8 x - 20$ using the AC method.

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Step 7.1

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 $- 20$ and whose sum is $8$ .

$- 2, 10$

Step 7.2

Write the factored form using these integers.

$- \frac{6 (x - 2)}{10 - x} \cdot \frac{x (10 + x) (10 - x)}{(x - 2) (x + 10)}$

Step 8

Simplify terms.

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Step 8.1

Cancel the common factor of $x - 2$ .

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Step 8.1.1

Move the leading negative in $- \frac{6 (x - 2)}{10 - x}$ into the numerator.

$\frac{- 6 (x - 2)}{10 - x} \cdot \frac{x (10 + x) (10 - x)}{(x - 2) (x + 10)}$

Step 8.1.2

Factor $x - 2$ out of $- 6 (x - 2)$ .

$\frac{(x - 2) \cdot - 6}{10 - x} \cdot \frac{x (10 + x) (10 - x)}{(x - 2) (x + 10)}$

Step 8.1.3

Cancel the common factor.

$\frac{(x - 2) \cdot - 6}{10 - x} \cdot \frac{x (10 + x) (10 - x)}{(x - 2) (x + 10)}$

Step 8.1.4

Rewrite the expression.

$\frac{- 6}{10 - x} \cdot \frac{x (10 + x) (10 - x)}{x + 10}$

Step 8.2

Cancel the common factor of $10 - x$ .

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Step 8.2.1

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

$\frac{- 6}{10 - x} \cdot \frac{(10 - x) (x (10 + x))}{x + 10}$

Step 8.2.2

Cancel the common factor.

$\frac{- 6}{10 - x} \cdot \frac{(10 - x) (x (10 + x))}{x + 10}$

Step 8.2.3

Rewrite the expression.

$- 6 \frac{x (10 + x)}{x + 10}$

Step 8.3

Combine $- 6$ and $\frac{x (10 + x)}{x + 10}$ .

$\frac{- 6 (x (10 + x))}{x + 10}$

Step 8.4

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

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Step 8.4.1

Reorder terms.

$\frac{- 6 (x (x + 10))}{x + 10}$

Step 8.4.2

Cancel the common factor.

$\frac{- 6 (x (x + 10))}{x + 10}$

Step 8.4.3

Divide $- 6 (x)$ by $1$ .

$- 6 (x)$

$- 6 x$