Algebra Examples

$\frac{2 x}{y^{2} - x^{2}} - \frac{x}{y - x}$

Step 1

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

$\frac{2 x}{(y + x) (y - x)} - \frac{x}{y - x}$

Step 2

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

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

Step 3

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

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Step 3.1

Multiply $\frac{x}{y - x}$ by $\frac{y + x}{y + x}$ .

$\frac{2 x}{(y + x) (y - x)} - \frac{x (y + x)}{(y - x) (y + x)}$

Step 3.2

Reorder the factors of $(y - x) (y + x)$ .

$\frac{2 x}{(y + x) (y - x)} - \frac{x (y + x)}{(y + x) (y - x)}$

Step 4

Combine the numerators over the common denominator.

$\frac{2 x - x (y + x)}{(y + x) (y - x)}$

Step 5

Simplify the numerator.

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Step 5.1

Factor $x$ out of $2 x - x (y + x)$ .

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Step 5.1.1

Factor $x$ out of $2 x$ .

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

Step 5.1.2

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

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

Step 5.1.3

Factor $x$ out of $x \cdot 2 + x (- (y + x))$ .

$\frac{x (2 - (y + x))}{(y + x) (y - x)}$

Step 5.2

Apply the distributive property.

$\frac{x (2 - y - x)}{(y + x) (y - x)}$