Pre-Algebra Examples

$\frac{6 x^{2} + 36 x y + 54}{x^{2} + 5 x y + 4 y^{2}} \div \frac{7 x + 21 y}{x + y}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{6 x^{2} + 36 x y + 54}{x^{2} + 5 x y + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 2

Factor $6$ out of $6 x^{2} + 36 x y + 54$ .

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Step 2.1

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

$\frac{6 (x^{2}) + 36 x y + 54}{x^{2} + 5 x y + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 2.2

Factor $6$ out of $36 x y$ .

$\frac{6 (x^{2}) + 6 (6 x y) + 54}{x^{2} + 5 x y + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 2.3

Factor $6$ out of $54$ .

$\frac{6 x^{2} + 6 (6 x y) + 6 \cdot 9}{x^{2} + 5 x y + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 2.4

Factor $6$ out of $6 x^{2} + 6 (6 x y)$ .

$\frac{6 (x^{2} + 6 x y) + 6 \cdot 9}{x^{2} + 5 x y + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 2.5

Factor $6$ out of $6 (x^{2} + 6 x y) + 6 \cdot 9$ .

$\frac{6 (x^{2} + 6 x y + 9)}{x^{2} + 5 x y + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 3

Factor by grouping.

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Step 3.1

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 $a \cdot c = 1 \cdot 4 = 4$ and whose sum is $b = 5$ .

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Step 3.1.1

Reorder terms.

$\frac{6 (x^{2} + 6 x y + 9)}{x^{2} + 4 y^{2} + 5 x y} \cdot \frac{x + y}{7 x + 21 y}$

Step 3.1.2

Reorder $4 y^{2}$ and $5 x y$ .

$\frac{6 (x^{2} + 6 x y + 9)}{x^{2} + 5 x y + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 3.1.3

Factor $5$ out of $5 x y$ .

$\frac{6 (x^{2} + 6 x y + 9)}{x^{2} + 5 (x y) + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 3.1.4

Rewrite $5$ as $1$ plus $4$

$\frac{6 (x^{2} + 6 x y + 9)}{x^{2} + (1 + 4) (x y) + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 3.1.5

Apply the distributive property.

$\frac{6 (x^{2} + 6 x y + 9)}{x^{2} + 1 (x y) + 4 (x y) + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 3.1.6

Multiply $x y$ by $1$ .

$\frac{6 (x^{2} + 6 x y + 9)}{x^{2} + x y + 4 (x y) + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 3.1.7

Move parentheses.

$\frac{6 (x^{2} + 6 x y + 9)}{x^{2} + x y + 4 x y + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{6 (x^{2} + 6 x y + 9)}{(x^{2} + x y) + 4 x y + 4 y^{2}} \cdot \frac{x + y}{7 x + 21 y}$

Step 3.2.2

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

$\frac{6 (x^{2} + 6 x y + 9)}{x (x + y) + 4 y (x + y)} \cdot \frac{x + y}{7 x + 21 y}$

Step 3.3

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

$\frac{6 (x^{2} + 6 x y + 9)}{(x + y) (x + 4 y)} \cdot \frac{x + y}{7 x + 21 y}$

Step 4

Cancel the common factor of $x + y$ .

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Step 4.1

Cancel the common factor.

$\frac{6 (x^{2} + 6 x y + 9)}{(x + y) (x + 4 y)} \cdot \frac{x + y}{7 x + 21 y}$

Step 4.2

Rewrite the expression.

$\frac{6 (x^{2} + 6 x y + 9)}{x + 4 y} \cdot \frac{1}{7 x + 21 y}$

Step 5

Multiply $\frac{6 (x^{2} + 6 x y + 9)}{x + 4 y}$ by $\frac{1}{7 x + 21 y}$ .

$\frac{6 (x^{2} + 6 x y + 9)}{(x + 4 y) (7 x + 21 y)}$

Step 6

Factor $7$ out of $7 x + 21 y$ .

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Step 6.1

Factor $7$ out of $7 x$ .

$\frac{6 (x^{2} + 6 x y + 9)}{(x + 4 y) (7 (x) + 21 y)}$

Step 6.2

Factor $7$ out of $21 y$ .

$\frac{6 (x^{2} + 6 x y + 9)}{(x + 4 y) (7 (x) + 7 (3 y))}$

Step 6.3

Factor $7$ out of $7 (x) + 7 (3 y)$ .

$\frac{6 (x^{2} + 6 x y + 9)}{(x + 4 y) (7 (x + 3 y))}$

$\frac{6 (x^{2} + 6 x y + 9)}{(x + 4 y) \cdot 7 (x + 3 y)}$

Step 7

Move $7$ to the left of $x + 4 y$ .

$\frac{6 (x^{2} + 6 x y + 9)}{7 (x + 4 y) (x + 3 y)}$

Step 8

Apply the distributive property.

$\frac{6 x^{2} + 6 (6 x y) + 6 \cdot 9}{7 (x + 4 y) (x + 3 y)}$

Step 9

Simplify.

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Step 9.1

Multiply $6$ by $6$ .

$\frac{6 x^{2} + 36 (x y) + 6 \cdot 9}{7 (x + 4 y) (x + 3 y)}$

Step 9.2

Multiply $6$ by $9$ .

$\frac{6 x^{2} + 36 (x y) + 54}{7 (x + 4 y) (x + 3 y)}$

$\frac{6 x^{2} + 36 x y + 54}{7 (x + 4 y) (x + 3 y)}$

Step 10

Split the fraction $\frac{6 x^{2} + 36 x y + 54}{7 (x + 4 y) (x + 3 y)}$ into two fractions.

$\frac{6 x^{2} + 36 x y}{7 (x + 4 y) (x + 3 y)} + \frac{54}{7 (x + 4 y) (x + 3 y)}$

Step 11

Split the fraction $\frac{6 x^{2} + 36 x y}{7 (x + 4 y) (x + 3 y)}$ into two fractions.

$\frac{6 x^{2}}{7 (x + 4 y) (x + 3 y)} + \frac{36 x y}{7 (x + 4 y) (x + 3 y)} + \frac{54}{7 (x + 4 y) (x + 3 y)}$