Pre-Algebra Examples

$\frac{x^{2} - y^{2}}{x^{2} + x y} \div \frac{3 x^{2} - 2 x y - y^{2}}{3 x^{2} + 6 x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{x^{2} - y^{2}}{x^{2} + x y} \cdot \frac{3 x^{2} + 6 x}{3 x^{2} - 2 x y - y^{2}}$

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 $b = y$ .

$\frac{(x + y) (x - y)}{x^{2} + x y} \cdot \frac{3 x^{2} + 6 x}{3 x^{2} - 2 x y - y^{2}}$

Step 3

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

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Step 3.1

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

$\frac{(x + y) (x - y)}{x \cdot x + x y} \cdot \frac{3 x^{2} + 6 x}{3 x^{2} - 2 x y - y^{2}}$

Step 3.2

Factor $x$ out of $x y$ .

$\frac{(x + y) (x - y)}{x \cdot x + x (y)} \cdot \frac{3 x^{2} + 6 x}{3 x^{2} - 2 x y - y^{2}}$

Step 3.3

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

$\frac{(x + y) (x - y)}{x (x + y)} \cdot \frac{3 x^{2} + 6 x}{3 x^{2} - 2 x y - y^{2}}$

Step 4

Cancel the common factor of $x + y$ .

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Step 4.1

Cancel the common factor.

$\frac{(x + y) (x - y)}{x (x + y)} \cdot \frac{3 x^{2} + 6 x}{3 x^{2} - 2 x y - y^{2}}$

Step 4.2

Rewrite the expression.

$\frac{x - y}{x} \cdot \frac{3 x^{2} + 6 x}{3 x^{2} - 2 x y - y^{2}}$

Step 5

Factor $3 x$ out of $3 x^{2} + 6 x$ .

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Step 5.1

Factor $3 x$ out of $3 x^{2}$ .

$\frac{x - y}{x} \cdot \frac{3 x (x) + 6 x}{3 x^{2} - 2 x y - y^{2}}$

Step 5.2

Factor $3 x$ out of $6 x$ .

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

Step 5.3

Factor $3 x$ out of $3 x (x) + 3 x (2)$ .

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

Step 6

Factor by grouping.

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Step 6.1

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 $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = - 2$ .

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Step 6.1.1

Reorder terms.

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

Step 6.1.2

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

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

Step 6.1.3

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

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

Step 6.1.4

Rewrite $- 2$ as $1$ plus $- 3$

$\frac{x - y}{x} \cdot \frac{3 x (x + 2)}{3 x^{2} + (1 - 3) (x y) - y^{2}}$

Step 6.1.5

Apply the distributive property.

$\frac{x - y}{x} \cdot \frac{3 x (x + 2)}{3 x^{2} + 1 (x y) - 3 (x y) - y^{2}}$

Step 6.1.6

Multiply $x y$ by $1$ .

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

Step 6.1.7

Move parentheses.

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

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

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

Step 6.2.2

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

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

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $3 x + y$ .

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

Step 7

Cancel the common factor of $x - y$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

$\frac{1}{x} \cdot \frac{3 x (x + 2)}{3 x + y}$

Step 8

Cancel the common factor of $x$ .

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Step 8.1

Factor $x$ out of $3 x (x + 2)$ .

$\frac{1}{x} \cdot \frac{x (3 (x + 2))}{3 x + y}$

Step 8.2

Cancel the common factor.

$\frac{1}{x} \cdot \frac{x (3 (x + 2))}{3 x + y}$

Step 8.3

Rewrite the expression.

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

Step 9

Apply the distributive property.

$\frac{3 x + 3 \cdot 2}{3 x + y}$

Step 10

Multiply $3$ by $2$ .

$\frac{3 x + 6}{3 x + y}$

Step 11

Split the fraction $\frac{3 x + 6}{3 x + y}$ into two fractions.

$\frac{3 x}{3 x + y} + \frac{6}{3 x + y}$