Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $x$ .

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Move the leading negative in $- \frac{5}{x}$ into the numerator.

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Multiply $x$ by $x$ .

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

Step 4.2

Move $- 2$ to the left of $x$ .

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

Step 4.3

Factor using the perfect square rule.

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Step 4.3.1

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

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

Step 4.3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 4.3.3

Rewrite the polynomial.

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

Step 4.3.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

Factor $5$ out of $- 5$ .

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

Step 5.1.3

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

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

Step 5.2

Cancel the common factor of ${(x - 1)}^{2}$ and $x - 1$ .

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Step 5.2.1

Factor $x - 1$ out of ${(x - 1)}^{2}$ .

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

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

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

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

Step 5.2.2.2

Cancel the common factor.

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

Step 5.2.2.3

Rewrite the expression.

$\frac{x - 1}{5}$