Basic Math Examples

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

Step 1

Move the negative in front of the fraction.

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

Step 2

To write $- \frac{x}{x - 5}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

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

Step 3

To write $\frac{x - 5}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

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

Step 4

Write each expression with a common denominator of $(x - 5) x$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{x}{x - 5}$ by $\frac{x}{x}$ .

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

Step 4.2

Multiply $\frac{x - 5}{x}$ by $\frac{x - 5}{x - 5}$ .

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

Step 4.3

Reorder the factors of $(x - 5) x$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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Step 6.1

Raise $x - 5$ to the power of $1$ .

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

Step 6.2

Raise $x - 5$ to the power of $1$ .

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

Step 6.3

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

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

Step 6.4

Add $1$ and $1$ .

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

Step 6.5

Rewrite $x \cdot x$ as $x^{2}$ .

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

Step 6.6

Reorder $- x^{2}$ and ${(x - 5)}^{2}$ .

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

Step 6.7

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

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

Step 6.8

Simplify.

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Step 6.8.1

Add $x$ and $x$ .

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

Step 6.8.2

Subtract $x$ from $x$ .

$\frac{(2 x - 5) (0 - 5)}{x (x - 5)}$

Step 6.8.3

Subtract $5$ from $0$ .

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

Step 7

Simplify the expression.

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Step 7.1

Move $- 5$ to the left of $2 x - 5$ .

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

Step 7.2

Move the negative in front of the fraction.

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