Algebra Examples

$\frac{1 - \frac{25}{x^{2}}}{1 + \frac{5}{x}}$

Step 1

Multiply the numerator and denominator of the fraction by $x^{2}$ .

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Step 1.1

Multiply $\frac{1 - \frac{25}{x^{2}}}{1 + \frac{5}{x}}$ by $\frac{x^{2}}{x^{2}}$ .

$\frac{x^{2}}{x^{2}} \cdot \frac{1 - \frac{25}{x^{2}}}{1 + \frac{5}{x}}$

Step 1.2

Combine.

$\frac{x^{2} (1 - \frac{25}{x^{2}})}{x^{2} (1 + \frac{5}{x})}$

Step 2

Apply the distributive property.

$\frac{x^{2} \cdot 1 + x^{2} (- \frac{25}{x^{2}})}{x^{2} \cdot 1 + x^{2} \frac{5}{x}}$

Step 3

Simplify by cancelling.

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Step 3.1

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

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Step 3.1.1

Move the leading negative in $- \frac{25}{x^{2}}$ into the numerator.

$\frac{x^{2} \cdot 1 + x^{2} \frac{- 25}{x^{2}}}{x^{2} \cdot 1 + x^{2} \frac{5}{x}}$

Step 3.1.2

Cancel the common factor.

$\frac{x^{2} \cdot 1 + x^{2} \frac{- 25}{x^{2}}}{x^{2} \cdot 1 + x^{2} \frac{5}{x}}$

Step 3.1.3

Rewrite the expression.

$\frac{x^{2} \cdot 1 - 25}{x^{2} \cdot 1 + x^{2} \frac{5}{x}}$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

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

$\frac{x^{2} \cdot 1 - 25}{x^{2} \cdot 1 + x \cdot x \frac{5}{x}}$

Step 3.2.2

Cancel the common factor.

$\frac{x^{2} \cdot 1 - 25}{x^{2} \cdot 1 + x \cdot x \frac{5}{x}}$

Step 3.2.3

Rewrite the expression.

$\frac{x^{2} \cdot 1 - 25}{x^{2} \cdot 1 + x \cdot 5}$

Step 4

Simplify the numerator.

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Step 4.1

Rewrite $x^{2} \cdot 1$ as ${(x \cdot 1)}^{2}$ .

$\frac{{(x \cdot 1)}^{2} - 25}{x^{2} \cdot 1 + x \cdot 5}$

Step 4.2

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

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

Step 4.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x \cdot 1$ and $b = 5$ .

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

Step 4.4

Simplify.

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Step 4.4.1

Multiply $x$ by $1$ .

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

Step 4.4.2

Multiply $x$ by $1$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $x$ out of $x^{2} \cdot 1 + x \cdot 5$ .

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Step 5.1.1

Factor $x$ out of $x^{2} \cdot 1$ .

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

Step 5.1.2

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

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

Step 5.1.3

Factor $x$ out of $x (x \cdot 1) + x (5)$ .

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

Step 5.2

Multiply $x$ by $1$ .

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

Step 6

Cancel the common factor of $x + 5$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{x - 5}{x}$