Pre-Algebra Examples

$\frac{x^{2} - 25}{x^{2}} \cdot \frac{x^{2} - 5 x}{x^{2} + 5 x - 50}$

Step 1

Simplify the numerator.

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Step 1.1

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

$\frac{x^{2} - 5^{2}}{x^{2}} \cdot \frac{x^{2} - 5 x}{x^{2} + 5 x - 50}$

Step 1.2

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 $b = 5$ .

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

Step 2

Factor $x$ out of $x^{2} - 5 x$ .

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Step 2.1

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

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

Step 2.2

Factor $x$ out of $- 5 x$ .

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

Step 2.3

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

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

Step 3

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

$- 5, 10$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Combine.

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

Step 4.2

Cancel the common factor of $x - 5$ .

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Step 4.2.1

Cancel the common factor.

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

Step 4.2.2

Rewrite the expression.

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

Step 4.3

Cancel the common factor of $x$ and $x^{2}$ .

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Step 4.3.1

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

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

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $x$ out of $x^{2} (x + 10)$ .

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

Step 4.3.2.2

Cancel the common factor.

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

Step 4.3.2.3

Rewrite the expression.

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