Pre-Algebra Examples

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

Step 1

To write $\frac{2 x}{y^{2} - x^{2}}$ as a fraction with a common denominator, multiply by $\frac{y - x}{y - x}$ .

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

Step 2

To write $- \frac{x}{y - x}$ as a fraction with a common denominator, multiply by $\frac{y^{2} - x^{2}}{y^{2} - x^{2}}$ .

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

Step 3

Write each expression with a common denominator of $(y^{2} - x^{2}) (y - x)$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.1

Multiply $\frac{2 x}{y^{2} - x^{2}}$ by $\frac{y - x}{y - x}$ .

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

Step 3.2

Multiply $\frac{x}{y - x}$ by $\frac{y^{2} - x^{2}}{y^{2} - x^{2}}$ .

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

Step 3.3

Reorder the factors of $(y^{2} - x^{2}) (y - x)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Rewrite $\frac{2 x (y - x) - x (y^{2} - x^{2})}{(y - x) (y^{2} - x^{2})}$ in a factored form.

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Step 5.1

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

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

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

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Multiply $- 1$ by $2$ .

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

Step 5.4

Apply the distributive property.

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

Step 5.5

Multiply $- - x^{2}$ .

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Step 5.5.1

Multiply $- 1$ by $- 1$ .

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

Step 5.5.2

Multiply $x^{2}$ by $1$ .

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

Step 5.6

Factor.

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Step 5.6.1

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 $b = x$ .

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

Step 5.6.2

Remove unnecessary parentheses.

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

Step 5.7

Combine exponents.

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Step 5.7.1

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

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

Step 5.7.2

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

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

Step 5.7.3

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

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

Step 5.7.4

Add $1$ and $1$ .

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