Algebra Examples

$\frac{20 x - 4 x^{2}}{2 x^{2} + 2 x} \cdot \frac{x^{2} - 100}{x + 10} \div \frac{x^{2} - 11 x + 10}{x^{2} - 1}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{20 x - 4 x^{2}}{2 x^{2} + 2 x} \cdot \frac{x^{2} - 100}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 2

Simplify with factoring out.

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Step 2.1

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

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Step 2.1.1

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

$\frac{4 x (5) - 4 x^{2}}{2 x^{2} + 2 x} \cdot \frac{x^{2} - 100}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 2.1.2

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

$\frac{4 x (5) + 4 x (- x)}{2 x^{2} + 2 x} \cdot \frac{x^{2} - 100}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 2.1.3

Factor $4 x$ out of $4 x (5) + 4 x (- x)$ .

$\frac{4 x (5 - x)}{2 x^{2} + 2 x} \cdot \frac{x^{2} - 100}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 2.2

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

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Step 2.2.1

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

$\frac{4 x (5 - x)}{2 x (x) + 2 x} \cdot \frac{x^{2} - 100}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 2.2.2

Factor $2 x$ out of $2 x$ .

$\frac{4 x (5 - x)}{2 x (x) + 2 x (1)} \cdot \frac{x^{2} - 100}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 2.2.3

Factor $2 x$ out of $2 x (x) + 2 x (1)$ .

$\frac{4 x (5 - x)}{2 x (x + 1)} \cdot \frac{x^{2} - 100}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{4 x (5 - x)}{2 x (x + 1)} \cdot \frac{x^{2} - 10^{2}}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

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 $b = 10$ .

$\frac{4 x (5 - x)}{2 x (x + 1)} \cdot \frac{(x + 10) (x - 10)}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 4

Simplify terms.

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Step 4.1

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

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Step 4.1.1

Factor $2$ out of $4 x (5 - x)$ .

$\frac{2 (2 x (5 - x))}{2 x (x + 1)} \cdot \frac{(x + 10) (x - 10)}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $2 x (x + 1)$ .

$\frac{2 (2 x (5 - x))}{2 (x (x + 1))} \cdot \frac{(x + 10) (x - 10)}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 4.1.2.2

Cancel the common factor.

$\frac{2 (2 x (5 - x))}{2 (x (x + 1))} \cdot \frac{(x + 10) (x - 10)}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 4.1.2.3

Rewrite the expression.

$\frac{2 x (5 - x)}{x (x + 1)} \cdot \frac{(x + 10) (x - 10)}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 4.2

Cancel the common factor of $x$ .

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Step 4.2.1

Cancel the common factor.

$\frac{2 x (5 - x)}{x (x + 1)} \cdot \frac{(x + 10) (x - 10)}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 4.2.2

Rewrite the expression.

$\frac{2 (5 - x)}{x + 1} \cdot \frac{(x + 10) (x - 10)}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 4.3

Cancel the common factor of $x + 10$ .

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Step 4.3.1

Cancel the common factor.

$\frac{2 (5 - x)}{x + 1} \cdot \frac{(x + 10) (x - 10)}{x + 10} \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 4.3.2

Divide $x - 10$ by $1$ .

$\frac{2 (5 - x)}{x + 1} \cdot (x - 10) \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 4.4

Multiply $\frac{2 (5 - x)}{x + 1}$ by $x - 10$ .

$\frac{2 (5 - x) (x - 10)}{x + 1} \cdot \frac{x^{2} - 1}{x^{2} - 11 x + 10}$

Step 5

Simplify the numerator.

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Step 5.1

Rewrite $1$ as $1^{2}$ .

$\frac{2 (5 - x) (x - 10)}{x + 1} \cdot \frac{x^{2} - 1^{2}}{x^{2} - 11 x + 10}$

Step 5.2

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 $b = 1$ .

$\frac{2 (5 - x) (x - 10)}{x + 1} \cdot \frac{(x + 1) (x - 1)}{x^{2} - 11 x + 10}$

Step 6

Factor $x^{2} - 11 x + 10$ using the AC method.

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Step 6.1

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 $10$ and whose sum is $- 11$ .

$- 10, - 1$

Step 6.2

Write the factored form using these integers.

$\frac{2 (5 - x) (x - 10)}{x + 1} \cdot \frac{(x + 1) (x - 1)}{(x - 10) (x - 1)}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $x - 10$ .

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Step 7.1.1

Factor $x - 10$ out of $2 (5 - x) (x - 10)$ .

$\frac{(x - 10) (2 (5 - x))}{x + 1} \cdot \frac{(x + 1) (x - 1)}{(x - 10) (x - 1)}$

Step 7.1.2

Cancel the common factor.

$\frac{(x - 10) (2 (5 - x))}{x + 1} \cdot \frac{(x + 1) (x - 1)}{(x - 10) (x - 1)}$

Step 7.1.3

Rewrite the expression.

$\frac{2 (5 - x)}{x + 1} \cdot \frac{(x + 1) (x - 1)}{x - 1}$

Step 7.2

Cancel the common factor of $x + 1$ .

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Step 7.2.1

Cancel the common factor.

$\frac{2 (5 - x)}{x + 1} \cdot \frac{(x + 1) (x - 1)}{x - 1}$

Step 7.2.2

Rewrite the expression.

$2 (5 - x) \frac{x - 1}{x - 1}$

Step 7.3

Combine $\frac{x - 1}{x - 1}$ and $2$ .

$\frac{(x - 1) \cdot 2}{x - 1} (5 - x)$

Step 7.4

Cancel the common factor of $x - 1$ .

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Step 7.4.1

Cancel the common factor.

$\frac{(x - 1) \cdot 2}{x - 1} (5 - x)$

Step 7.4.2

Divide $2$ by $1$ .

$2 (5 - x)$

Step 7.5

Apply the distributive property.

$2 \cdot 5 + 2 (- x)$

Step 7.6

Multiply.

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Step 7.6.1

Multiply $2$ by $5$ .

$10 + 2 (- x)$

Step 7.6.2

Multiply $- 1$ by $2$ .

$10 - 2 x$