Algebra Examples

$\frac{2 x}{x^{2} - 5 x - 50} \cdot \frac{x^{2} - 25}{x + 10} \cdot \frac{600 x - 6 x^{3}}{5 - x}$

Step 1

Simplify the numerator.

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Step 1.1

Factor $6 x$ out of $600 x - 6 x^{3}$ .

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Step 1.1.1

Factor $6 x$ out of $600 x$ .

$\frac{2 x}{x^{2} - 5 x - 50} \cdot \frac{x^{2} - 25}{x + 10} \cdot \frac{6 x (100) - 6 x^{3}}{5 - x}$

Step 1.1.2

Factor $6 x$ out of $- 6 x^{3}$ .

$\frac{2 x}{x^{2} - 5 x - 50} \cdot \frac{x^{2} - 25}{x + 10} \cdot \frac{6 x (100) + 6 x (- x^{2})}{5 - x}$

Step 1.1.3

Factor $6 x$ out of $6 x (100) + 6 x (- x^{2})$ .

$\frac{2 x}{x^{2} - 5 x - 50} \cdot \frac{x^{2} - 25}{x + 10} \cdot \frac{6 x (100 - x^{2})}{5 - x}$

Step 1.2

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

$\frac{2 x}{x^{2} - 5 x - 50} \cdot \frac{x^{2} - 25}{x + 10} \cdot \frac{6 x (10^{2} - x^{2})}{5 - x}$

Step 1.3

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 $b = x$ .

$\frac{2 x}{x^{2} - 5 x - 50} \cdot \frac{x^{2} - 25}{x + 10} \cdot \frac{6 x (10 + x) (10 - x)}{5 - x}$

Step 2

Factor $x^{2} - 5 x - 50$ 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 $- 50$ and whose sum is $- 5$ .

$- 10, 5$

Step 2.2

Write the factored form using these integers.

$\frac{2 x}{(x - 10) (x + 5)} \cdot \frac{x^{2} - 25}{x + 10} \cdot \frac{6 x (10 + x) (10 - x)}{5 - x}$

Step 3

Simplify the numerator.

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Step 3.1

Rewrite $25$ as $5^{2}$ .

$\frac{2 x}{(x - 10) (x + 5)} \cdot \frac{x^{2} - 5^{2}}{x + 10} \cdot \frac{6 x (10 + x) (10 - x)}{5 - x}$

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 $b = 5$ .

$\frac{2 x}{(x - 10) (x + 5)} \cdot \frac{(x + 5) (x - 5)}{x + 10} \cdot \frac{6 x (10 + x) (10 - x)}{5 - x}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x + 5$ .

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Step 4.1.1

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

$\frac{2 x}{(x + 5) (x - 10)} \cdot \frac{(x + 5) (x - 5)}{x + 10} \cdot \frac{6 x (10 + x) (10 - x)}{5 - x}$

Step 4.1.2

Cancel the common factor.

$\frac{2 x}{(x + 5) (x - 10)} \cdot \frac{(x + 5) (x - 5)}{x + 10} \cdot \frac{6 x (10 + x) (10 - x)}{5 - x}$

Step 4.1.3

Rewrite the expression.

$\frac{2 x}{x - 10} \cdot \frac{x - 5}{x + 10} \cdot \frac{6 x (10 + x) (10 - x)}{5 - x}$

Step 4.2

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

$\frac{2 x (x - 5)}{(x - 10) (x + 10)} \cdot \frac{6 x (10 + x) (10 - x)}{5 - x}$

Step 5

Multiply $\frac{2 x (x - 5)}{(x - 10) (x + 10)} \cdot \frac{6 x (10 + x) (10 - x)}{5 - x}$ .

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Step 5.1

Multiply $\frac{2 x (x - 5)}{(x - 10) (x + 10)}$ by $\frac{6 x (10 + x) (10 - x)}{5 - x}$ .

$\frac{2 x (x - 5) (6 x (10 + x) (10 - x))}{(x - 10) (x + 10) (5 - x)}$

Step 5.2

Multiply $6$ by $2$ .

$\frac{12 x (x - 5) ((x (10 + x)) (10 - x))}{(x - 10) (x + 10) (5 - x)}$

Step 5.3

Raise $x$ to the power of $1$ .

$\frac{12 (x^{1} x) (x - 5) ((10 + x) (10 - x))}{(x - 10) (x + 10) (5 - x)}$

Step 5.4

Raise $x$ to the power of $1$ .

$\frac{12 (x^{1} x^{1}) (x - 5) ((10 + x) (10 - x))}{(x - 10) (x + 10) (5 - x)}$

Step 5.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{12 x^{1 + 1} (x - 5) ((10 + x) (10 - x))}{(x - 10) (x + 10) (5 - x)}$

Step 5.6

Add $1$ and $1$ .

$\frac{12 x^{2} (x - 5) ((10 + x) (10 - x))}{(x - 10) (x + 10) (5 - x)}$

Step 6

Cancel the common factor of $x - 5$ and $5 - x$ .

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Step 6.1

Factor $- 1$ out of $x$ .

$\frac{12 x^{2} (- 1 (- x) - 5) ((10 + x) (10 - x))}{(x - 10) (x + 10) (5 - x)}$

Step 6.2

Rewrite $- 5$ as $- 1 (5)$ .

$\frac{12 x^{2} (- 1 (- x) - 1 (5)) ((10 + x) (10 - x))}{(x - 10) (x + 10) (5 - x)}$

Step 6.3

Factor $- 1$ out of $- 1 (- x) - 1 (5)$ .

$\frac{12 x^{2} (- 1 (- x + 5)) ((10 + x) (10 - x))}{(x - 10) (x + 10) (5 - x)}$

Step 6.4

Reorder terms.

$\frac{12 x^{2} (- 1 (- x + 5)) ((10 + x) (10 - x))}{(x - 10) (x + 10) (- x + 5)}$

Step 6.5

Cancel the common factor.

$\frac{12 x^{2} (- 1 (- x + 5)) ((10 + x) (10 - x))}{(x - 10) (x + 10) (- x + 5)}$

Step 6.6

Rewrite the expression.

$\frac{12 x^{2} \cdot (- 1) ((10 + x) (10 - x))}{(x - 10) (x + 10)}$

Step 7

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

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Step 7.1

Reorder terms.

$\frac{12 x^{2} \cdot (- 1) ((x + 10) (10 - x))}{(x - 10) (x + 10)}$

Step 7.2

Cancel the common factor.

$\frac{12 x^{2} \cdot (- 1) ((x + 10) (10 - x))}{(x - 10) (x + 10)}$

Step 7.3

Rewrite the expression.

$\frac{12 x^{2} \cdot (- 1) (10 - x)}{x - 10}$

Step 8

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

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Step 8.1

Rewrite $10$ as $- 1 (- 10)$ .

$\frac{12 x^{2} \cdot (- 1) (- 1 (- 10) - x)}{x - 10}$

Step 8.2

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

$\frac{12 x^{2} \cdot (- 1) (- 1 (- 10) - (x))}{x - 10}$

Step 8.3

Factor $- 1$ out of $- 1 (- 10) - (x)$ .

$\frac{12 x^{2} \cdot (- 1) (- 1 (- 10 + x))}{x - 10}$

Step 8.4

Reorder terms.

$\frac{12 x^{2} \cdot - 1 (- 1 (x - 10))}{x - 10}$

Step 8.5

Cancel the common factor.

$\frac{12 x^{2} \cdot - 1 (- 1 (x - 10))}{x - 10}$

Step 8.6

Divide $12 x^{2} \cdot - 1 \cdot (- 1)$ by $1$ .

$12 x^{2} \cdot - 1 \cdot (- 1)$

Step 9

Multiply.

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Step 9.1

Multiply $- 1$ by $12$ .

$- 12 x^{2} \cdot - 1$

Step 9.2

Multiply $- 1$ by $- 12$ .

$12 x^{2}$