Pre-Algebra Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

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 $b = y$ .

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

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

Factor $x$ out of $x^{2} \cdot 2 y$ .

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

Step 2.1.2

Factor $x$ out of $2 x$ .

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

Step 2.1.3

Factor $x$ out of $y x$ .

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

Step 2.1.4

Factor $x$ out of $x (x \cdot 2 y) + x \cdot 2$ .

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

Step 2.1.5

Factor $x$ out of $x (x \cdot 2 y + 2) + x y$ .

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

Step 2.2

Move $2$ to the left of $x$ .

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

Step 3

Apply the distributive property.

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

Step 4

Expand $(3 x + 3 y) (x - y)$ using the FOIL Method.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Apply the distributive property.

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

Step 4.3

Apply the distributive property.

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

Step 5

Simplify and combine like terms.

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Step 5.1

Simplify each term.

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Step 5.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1.1.1

Move $x$ .

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

Step 5.1.1.2

Multiply $x$ by $x$ .

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

Step 5.1.2

Rewrite using the commutative property of multiplication.

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

Step 5.1.3

Multiply $3$ by $- 1$ .

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

Step 5.1.4

Rewrite using the commutative property of multiplication.

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

Step 5.1.5

Multiply $y$ by $y$ by adding the exponents.

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Step 5.1.5.1

Move $y$ .

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

Step 5.1.5.2

Multiply $y$ by $y$ .

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

Step 5.1.6

Multiply $3$ by $- 1$ .

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

Step 5.2

Add $- 3 x y$ and $3 y x$ .

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Step 5.2.1

Move $y$ .

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

Step 5.2.2

Add $- 3 x y$ and $3 x y$ .

$\frac{3 x^{2} + 0 - 3 y^{2}}{x (2 x y + 2 + y)}$

Step 5.3

Add $3 x^{2}$ and $0$ .

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

Step 6

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

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

Step 7

Cancel the common factor of $x^{2}$ and $x$ .

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Step 7.1

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

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

Step 7.2

Cancel the common factors.

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Step 7.2.1

Cancel the common factor.

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

Step 7.2.2

Rewrite the expression.

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

Step 8

Move the negative in front of the fraction.

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