Pre-Algebra Examples

$\frac{4 x - 12 y}{x^{2} - 4 y^{2}} \div \frac{x - 3 y}{x^{2} - 7 x y - 18 y^{2}}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{4 x - 12 y}{x^{2} - 4 y^{2}} \cdot \frac{x^{2} - 7 x y - 18 y^{2}}{x - 3 y}$

Step 2

Factor $4$ out of $4 x - 12 y$ .

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Step 2.1

Factor $4$ out of $4 x$ .

$\frac{4 (x) - 12 y}{x^{2} - 4 y^{2}} \cdot \frac{x^{2} - 7 x y - 18 y^{2}}{x - 3 y}$

Step 2.2

Factor $4$ out of $- 12 y$ .

$\frac{4 (x) + 4 (- 3 y)}{x^{2} - 4 y^{2}} \cdot \frac{x^{2} - 7 x y - 18 y^{2}}{x - 3 y}$

Step 2.3

Factor $4$ out of $4 (x) + 4 (- 3 y)$ .

$\frac{4 (x - 3 y)}{x^{2} - 4 y^{2}} \cdot \frac{x^{2} - 7 x y - 18 y^{2}}{x - 3 y}$

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $4 y^{2}$ as ${(2 y)}^{2}$ .

$\frac{4 (x - 3 y)}{x^{2} - {(2 y)}^{2}} \cdot \frac{x^{2} - 7 x y - 18 y^{2}}{x - 3 y}$

Step 3.2

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 $b = 2 y$ .

$\frac{4 (x - 3 y)}{(x + 2 y) (x - (2 y))} \cdot \frac{x^{2} - 7 x y - 18 y^{2}}{x - 3 y}$

Step 3.3

Multiply $2$ by $- 1$ .

$\frac{4 (x - 3 y)}{(x + 2 y) (x - 2 y)} \cdot \frac{x^{2} - 7 x y - 18 y^{2}}{x - 3 y}$

Step 4

Simplify terms.

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Step 4.1

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

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Step 4.1.1

Factor $x - 3 y$ out of $4 (x - 3 y)$ .

$\frac{(x - 3 y) \cdot 4}{(x + 2 y) (x - 2 y)} \cdot \frac{x^{2} - 7 x y - 18 y^{2}}{x - 3 y}$

Step 4.1.2

Cancel the common factor.

$\frac{(x - 3 y) \cdot 4}{(x + 2 y) (x - 2 y)} \cdot \frac{x^{2} - 7 x y - 18 y^{2}}{x - 3 y}$

Step 4.1.3

Rewrite the expression.

$\frac{4}{(x + 2 y) (x - 2 y)} (x^{2} - 7 x y - 18 y^{2})$

Step 4.2

Multiply $\frac{4}{(x + 2 y) (x - 2 y)}$ by $x^{2} - 7 x y - 18 y^{2}$ .

$\frac{4 (x^{2} - 7 x y - 18 y^{2})}{(x + 2 y) (x - 2 y)}$

Step 5

Factor by grouping.

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Step 5.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 1 \cdot - 18 = - 18$ and whose sum is $b = - 7$ .

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Step 5.1.1

Reorder terms.

$\frac{4 (x^{2} - 18 y^{2} - 7 x y)}{(x + 2 y) (x - 2 y)}$

Step 5.1.2

Reorder $- 18 y^{2}$ and $- 7 x y$ .

$\frac{4 (x^{2} - 7 x y - 18 y^{2})}{(x + 2 y) (x - 2 y)}$

Step 5.1.3

Factor $- 7$ out of $- 7 x y$ .

$\frac{4 (x^{2} - 7 (x y) - 18 y^{2})}{(x + 2 y) (x - 2 y)}$

Step 5.1.4

Rewrite $- 7$ as $2$ plus $- 9$

$\frac{4 (x^{2} + (2 - 9) (x y) - 18 y^{2})}{(x + 2 y) (x - 2 y)}$

Step 5.1.5

Apply the distributive property.

$\frac{4 (x^{2} + 2 (x y) - 9 (x y) - 18 y^{2})}{(x + 2 y) (x - 2 y)}$

Step 5.1.6

Move parentheses.

$\frac{4 (x^{2} + 2 x y - 9 x y - 18 y^{2})}{(x + 2 y) (x - 2 y)}$

Step 5.2

Factor out the greatest common factor from each group.

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Step 5.2.1

Group the first two terms and the last two terms.

$\frac{4 ((x^{2} + 2 x y) - 9 x y - 18 y^{2})}{(x + 2 y) (x - 2 y)}$

Step 5.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{4 (x (x + 2 y) - 9 y (x + 2 y))}{(x + 2 y) (x - 2 y)}$

Step 5.3

Factor the polynomial by factoring out the greatest common factor, $x + 2 y$ .

$\frac{4 ((x + 2 y) (x - 9 y))}{(x + 2 y) (x - 2 y)}$

$\frac{4 (x + 2 y) (x - 9 y)}{(x + 2 y) (x - 2 y)}$

Step 6

Cancel the common factor of $x + 2 y$ .

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Step 6.1

Cancel the common factor.

$\frac{4 (x + 2 y) (x - 9 y)}{(x + 2 y) (x - 2 y)}$

Step 6.2

Rewrite the expression.

$\frac{4 (x - 9 y)}{x - 2 y}$