Precalculus Examples

$\frac{4 x^{2} - 9 y^{2}}{6 x^{2} - 9 x y} \cdot \frac{6 x y}{4 x y + 6 y^{2}}$

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

Rewrite $9 y^{2}$ as ${(3 y)}^{2}$ .

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

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x$ and $b = 3 y$ .

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

Step 1.4

Multiply $3$ by $- 1$ .

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

Step 2

Simplify terms.

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Step 2.1

Factor $3 x$ out of $6 x^{2} - 9 x y$ .

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Step 2.1.1

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

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

Step 2.1.2

Factor $3 x$ out of $- 9 x y$ .

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

Step 2.1.3

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

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

Step 2.2

Factor $2 y$ out of $4 x y + 6 y^{2}$ .

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Step 2.2.1

Factor $2 y$ out of $4 x y$ .

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

Cancel the common factor of $2 x + 3 y$ .

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Step 2.3.1

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

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

Step 2.3.2

Cancel the common factor.

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

Step 2.3.3

Rewrite the expression.

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

Step 2.4

Cancel the common factor of $3 x$ .

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Step 2.4.1

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

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

Step 2.4.2

Cancel the common factor.

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

Step 2.4.3

Rewrite the expression.

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

Step 2.5

Multiply $\frac{2 x - 3 y}{2 x - 3 y}$ by $\frac{2 y}{2 y}$ .

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

Step 2.6

Cancel the common factor of $2 x - 3 y$ .

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Step 2.6.1

Cancel the common factor.

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

Step 2.6.2

Rewrite the expression.

$\frac{2 y}{2 y}$

Step 2.7

Cancel the common factor of $2$ .

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Step 2.7.1

Cancel the common factor.

$\frac{2 y}{2 y}$

Step 2.7.2

Rewrite the expression.

$\frac{y}{y}$

Step 2.8

Cancel the common factor of $y$ .

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Step 2.8.1

Cancel the common factor.

$\frac{y}{y}$

Step 2.8.2

Rewrite the expression.

$1$