Algebra Examples

$\frac{x + 1}{x + 1} \cdot \frac{x^{2} - 1}{4 x^{2} - 40 x} \div \frac{x^{2} + 9 x - 10}{x^{2} - 100}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x + 1}{x + 1} \cdot \frac{x^{2} - 1}{4 x^{2} - 40 x} \frac{x^{2} - 100}{x^{2} + 9 x - 10}$

Step 2

Reduce the expression by cancelling the common factors.

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Step 2.1

Cancel the common factor of $x + 1$ .

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Step 2.1.1

Cancel the common factor.

$\frac{x + 1}{x + 1} \cdot \frac{x^{2} - 1}{4 x^{2} - 40 x} \frac{x^{2} - 100}{x^{2} + 9 x - 10}$

Step 2.1.2

Rewrite the expression.

$1 \cdot \frac{x^{2} - 1}{4 x^{2} - 40 x} \frac{x^{2} - 100}{x^{2} + 9 x - 10}$

Step 2.2

Multiply $\frac{x^{2} - 1}{4 x^{2} - 40 x}$ by $1$ .

$\frac{x^{2} - 1}{4 x^{2} - 40 x} \cdot \frac{x^{2} - 100}{x^{2} + 9 x - 10}$

Step 3

Simplify the numerator.

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Step 3.1

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

$\frac{x^{2} - 1^{2}}{4 x^{2} - 40 x} \cdot \frac{x^{2} - 100}{x^{2} + 9 x - 10}$

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 $b = 1$ .

$\frac{(x + 1) (x - 1)}{4 x^{2} - 40 x} \cdot \frac{x^{2} - 100}{x^{2} + 9 x - 10}$

Step 4

Factor $4 x$ out of $4 x^{2} - 40 x$ .

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Step 4.1

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

$\frac{(x + 1) (x - 1)}{4 x (x) - 40 x} \cdot \frac{x^{2} - 100}{x^{2} + 9 x - 10}$

Step 4.2

Factor $4 x$ out of $- 40 x$ .

$\frac{(x + 1) (x - 1)}{4 x (x) + 4 x (- 10)} \cdot \frac{x^{2} - 100}{x^{2} + 9 x - 10}$

Step 4.3

Factor $4 x$ out of $4 x (x) + 4 x (- 10)$ .

$\frac{(x + 1) (x - 1)}{4 x (x - 10)} \cdot \frac{x^{2} - 100}{x^{2} + 9 x - 10}$

Step 5

Simplify the numerator.

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Step 5.1

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

$\frac{(x + 1) (x - 1)}{4 x (x - 10)} \cdot \frac{x^{2} - 10^{2}}{x^{2} + 9 x - 10}$

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 $b = 10$ .

$\frac{(x + 1) (x - 1)}{4 x (x - 10)} \cdot \frac{(x + 10) (x - 10)}{x^{2} + 9 x - 10}$

Step 6

Factor $x^{2} + 9 x - 10$ 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 $9$ .

$- 1, 10$

Step 6.2

Write the factored form using these integers.

$\frac{(x + 1) (x - 1)}{4 x (x - 10)} \cdot \frac{(x + 10) (x - 10)}{(x - 1) (x + 10)}$

Step 7

Simplify terms.

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Step 7.1

Cancel the common factor of $x - 1$ .

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Step 7.1.1

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

$\frac{(x - 1) (x + 1)}{4 x (x - 10)} \cdot \frac{(x + 10) (x - 10)}{(x - 1) (x + 10)}$

Step 7.1.2

Cancel the common factor.

$\frac{(x - 1) (x + 1)}{4 x (x - 10)} \cdot \frac{(x + 10) (x - 10)}{(x - 1) (x + 10)}$

Step 7.1.3

Rewrite the expression.

$\frac{x + 1}{4 x (x - 10)} \cdot \frac{(x + 10) (x - 10)}{x + 10}$

Step 7.2

Cancel the common factor of $x - 10$ .

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Step 7.2.1

Factor $x - 10$ out of $4 x (x - 10)$ .

$\frac{x + 1}{(x - 10) (4 x)} \cdot \frac{(x + 10) (x - 10)}{x + 10}$

Step 7.2.2

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

$\frac{x + 1}{(x - 10) (4 x)} \cdot \frac{(x - 10) (x + 10)}{x + 10}$

Step 7.2.3

Cancel the common factor.

$\frac{x + 1}{(x - 10) (4 x)} \cdot \frac{(x - 10) (x + 10)}{x + 10}$

Step 7.2.4

Rewrite the expression.

$\frac{x + 1}{4 x} \cdot \frac{x + 10}{x + 10}$

Step 7.3

Multiply $\frac{x + 1}{4 x}$ by $\frac{x + 10}{x + 10}$ .

$\frac{(x + 1) (x + 10)}{4 x (x + 10)}$

Step 7.4

Cancel the common factor of $x + 10$ .

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Step 7.4.1

Cancel the common factor.

$\frac{(x + 1) (x + 10)}{4 x (x + 10)}$

Step 7.4.2

Rewrite the expression.

$\frac{x + 1}{4 x}$