Algebra Examples

$\frac{\frac{x - 10}{25} + \frac{1}{x}}{1 - \frac{x}{5}}$

Step 1

Multiply the numerator and denominator of the fraction by $25 x$ .

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Step 1.1

Multiply $\frac{\frac{x - 10}{25} + \frac{1}{x}}{1 - \frac{x}{5}}$ by $\frac{25 x}{25 x}$ .

$\frac{25 x}{25 x} \cdot \frac{\frac{x - 10}{25} + \frac{1}{x}}{1 - \frac{x}{5}}$

Step 1.2

Combine.

$\frac{25 x (\frac{x - 10}{25} + \frac{1}{x})}{25 x (1 - \frac{x}{5})}$

Step 2

Apply the distributive property.

$\frac{25 x \frac{x - 10}{25} + 25 x \frac{1}{x}}{25 x \cdot 1 + 25 x (- \frac{x}{5})}$

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $25$ .

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Step 3.1.1

Factor $25$ out of $25 x$ .

$\frac{25 (x) \frac{x - 10}{25} + 25 x \frac{1}{x}}{25 x \cdot 1 + 25 x (- \frac{x}{5})}$

Step 3.1.2

Cancel the common factor.

$\frac{25 x \frac{x - 10}{25} + 25 x \frac{1}{x}}{25 x \cdot 1 + 25 x (- \frac{x}{5})}$

Step 3.1.3

Rewrite the expression.

$\frac{x (x - 10) + 25 x \frac{1}{x}}{25 x \cdot 1 + 25 x (- \frac{x}{5})}$

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Factor $x$ out of $25 x$ .

$\frac{x (x - 10) + x \cdot 25 \frac{1}{x}}{25 x \cdot 1 + 25 x (- \frac{x}{5})}$

Step 3.2.2

Cancel the common factor.

$\frac{x (x - 10) + x \cdot 25 \frac{1}{x}}{25 x \cdot 1 + 25 x (- \frac{x}{5})}$

Step 3.2.3

Rewrite the expression.

$\frac{x (x - 10) + 25}{25 x \cdot 1 + 25 x (- \frac{x}{5})}$

Step 3.3

Cancel the common factor of $5$ .

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Step 3.3.1

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

$\frac{x (x - 10) + 25}{25 x \cdot 1 + 25 x \frac{- x}{5}}$

Step 3.3.2

Factor $5$ out of $25 x$ .

$\frac{x (x - 10) + 25}{25 x \cdot 1 + 5 (5 x) \frac{- x}{5}}$

Step 3.3.3

Cancel the common factor.

$\frac{x (x - 10) + 25}{25 x \cdot 1 + 5 (5 x) \frac{- x}{5}}$

Step 3.3.4

Rewrite the expression.

$\frac{x (x - 10) + 25}{25 x \cdot 1 + 5 x (- x)}$

Step 3.4

Multiply $- 1$ by $5$ .

$\frac{x (x - 10) + 25}{25 x \cdot 1 - 5 x \cdot x}$

Step 3.5

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

$\frac{x (x - 10) + 25}{25 x \cdot 1 - 5 (x^{1} x)}$

Step 3.6

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

$\frac{x (x - 10) + 25}{25 x \cdot 1 - 5 (x^{1} x^{1})}$

Step 3.7

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

$\frac{x (x - 10) + 25}{25 x \cdot 1 - 5 x^{1 + 1}}$

Step 3.8

Add $1$ and $1$ .

$\frac{x (x - 10) + 25}{25 x \cdot 1 - 5 x^{2}}$

Step 4

Simplify the numerator.

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Step 4.1

Apply the distributive property.

$\frac{x \cdot x + x \cdot - 10 + 25}{25 x \cdot 1 - 5 x^{2}}$

Step 4.2

Multiply $x$ by $x$ .

$\frac{x^{2} + x \cdot - 10 + 25}{25 x \cdot 1 - 5 x^{2}}$

Step 4.3

Move $- 10$ to the left of $x$ .

$\frac{x^{2} - 10 \cdot x + 25}{25 x \cdot 1 - 5 x^{2}}$

Step 4.4

Factor using the perfect square rule.

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Step 4.4.1

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

$\frac{x^{2} - 10 x + 5^{2}}{25 x \cdot 1 - 5 x^{2}}$

Step 4.4.2

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

$10 x = 2 \cdot x \cdot 5$

Step 4.4.3

Rewrite the polynomial.

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

Step 4.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 5$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $5 x$ out of $25 x \cdot 1 - 5 x^{2}$ .

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Step 5.1.1

Factor $5 x$ out of $25 x \cdot 1$ .

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Multiply $5$ by $1$ .

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

Step 6

Simplify terms.

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Step 6.1

Cancel the common factor of ${(x - 5)}^{2}$ and $5 - x$ .

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Step 6.1.1

Factor $- 1$ out of $x$ .

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

Apply the product rule to $- 1 (- x + 5)$ .

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

Step 6.1.5

Raise $- 1$ to the power of $2$ .

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

Step 6.1.6

Multiply ${(- x + 5)}^{2}$ by $1$ .

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

Step 6.1.7

Reorder terms.

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

Step 6.1.8

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

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

Step 6.1.9

Cancel the common factors.

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Step 6.1.9.1

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

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

Step 6.1.9.2

Cancel the common factor.

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

Step 6.1.9.3

Rewrite the expression.

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

Step 6.2

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

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Simplify the expression.

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Step 6.5.1

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

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

Step 6.5.2

Move the negative in front of the fraction.

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