Algebra Examples

$\frac{x^{2} - y^{2}}{4 x + 4 y} \div \frac{3 y - 3 x}{12 x^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - y^{2}}{4 x + 4 y} \cdot \frac{12 x^{2}}{3 y - 3 x}$

Step 2

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

$\frac{(x + y) (x - y)}{4 x + 4 y} \cdot \frac{12 x^{2}}{3 y - 3 x}$

Step 3

Simplify terms.

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Step 3.1

Factor $4$ out of $4 x + 4 y$ .

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Step 3.1.1

Factor $4$ out of $4 x$ .

$\frac{(x + y) (x - y)}{4 (x) + 4 y} \cdot \frac{12 x^{2}}{3 y - 3 x}$

Step 3.1.2

Factor $4$ out of $4 y$ .

$\frac{(x + y) (x - y)}{4 (x) + 4 (y)} \cdot \frac{12 x^{2}}{3 y - 3 x}$

Step 3.1.3

Factor $4$ out of $4 (x) + 4 (y)$ .

$\frac{(x + y) (x - y)}{4 (x + y)} \cdot \frac{12 x^{2}}{3 y - 3 x}$

Step 3.2

Cancel the common factor of $4$ .

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Step 3.2.1

Factor $4$ out of $12 x^{2}$ .

$\frac{(x + y) (x - y)}{4 (x + y)} \cdot \frac{4 (3 x^{2})}{3 y - 3 x}$

Step 3.2.2

Cancel the common factor.

$\frac{(x + y) (x - y)}{4 (x + y)} \cdot \frac{4 (3 x^{2})}{3 y - 3 x}$

Step 3.2.3

Rewrite the expression.

$\frac{(x + y) (x - y)}{x + y} \cdot \frac{3 x^{2}}{3 y - 3 x}$

Step 3.3

Multiply $\frac{(x + y) (x - y)}{x + y}$ by $\frac{3 x^{2}}{3 y - 3 x}$ .

$\frac{(x + y) (x - y) (3 x^{2})}{(x + y) (3 y - 3 x)}$

Step 3.4

Cancel the common factor of $x + y$ .

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Step 3.4.1

Cancel the common factor.

$\frac{(x + y) (x - y) (3 x^{2})}{(x + y) (3 y - 3 x)}$

Step 3.4.2

Rewrite the expression.

$\frac{(x - y) (3 x^{2})}{3 y - 3 x}$

Step 3.5

Cancel the common factor of $3$ and $3 y - 3 x$ .

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Step 3.5.1

Factor $3$ out of $(x - y) (3 x^{2})$ .

$\frac{3 ((x - y) (x^{2}))}{3 y - 3 x}$

Step 3.5.2

Cancel the common factors.

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Step 3.5.2.1

Factor $3$ out of $3 y$ .

$\frac{3 ((x - y) (x^{2}))}{3 (y) - 3 x}$

Step 3.5.2.2

Factor $3$ out of $- 3 x$ .

$\frac{3 ((x - y) (x^{2}))}{3 (y) + 3 (- x)}$

Step 3.5.2.3

Factor $3$ out of $3 (y) + 3 (- x)$ .

$\frac{3 ((x - y) (x^{2}))}{3 (y - x)}$

Step 3.5.2.4

Cancel the common factor.

$\frac{3 ((x - y) (x^{2}))}{3 (y - x)}$

Step 3.5.2.5

Rewrite the expression.

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

Step 3.6

Cancel the common factor of $x - y$ and $y - x$ .

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Step 3.6.1

Factor $- 1$ out of $x$ .

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

Step 3.6.2

Factor $- 1$ out of $- y$ .

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

Step 3.6.3

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

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

Step 3.6.4

Reorder terms.

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

Step 3.6.5

Cancel the common factor.

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

Step 3.6.6

Divide $- 1 x^{2}$ by $1$ .

$- 1 x^{2}$

Step 3.7

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

$- x^{2}$