Factor over the Complex Numbers ((4x^2-y^2)/(xy))÷(2/y)-1/x

$\frac{4 x^{2} - y^{2}}{x y} \div \frac{2}{y} - \frac{1}{x}$

Step 1

To write $\frac{4 x^{2} - y^{2}}{x y} \div \frac{2}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{4 x^{2} - y^{2}}{x y} \div \frac{2}{y} \cdot \frac{x}{x} - \frac{1}{x}$

Step 2

To write $- \frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{\frac{2}{y}}{\frac{2}{y}}$ .

$\frac{4 x^{2} - y^{2}}{x y} \div \frac{2}{y} \cdot \frac{x}{x} - \frac{1}{x} \cdot \frac{\frac{2}{y}}{\frac{2}{y}}$

Step 3

Write each expression with a common denominator of $\frac{2}{y} x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{4 x^{2} - y^{2}}{x y} \div \frac{2}{y}$ by $\frac{x}{x}$ .

$\frac{\frac{4 x^{2} - y^{2}}{x y} x}{\frac{2}{y} x} - \frac{1}{x} \cdot \frac{\frac{2}{y}}{\frac{2}{y}}$

Combine $\frac{2}{y}$ and $x$ .

$\frac{\frac{4 x^{2} - y^{2}}{x y} x}{\frac{2 x}{y}} - \frac{1}{x} \cdot \frac{\frac{2}{y}}{\frac{2}{y}}$

Multiply $\frac{1}{x}$ by $\frac{\frac{2}{y}}{\frac{2}{y}}$ .

$\frac{\frac{4 x^{2} - y^{2}}{x y} x}{\frac{2 x}{y}} - \frac{\frac{2}{y}}{x \frac{2}{y}}$

Combine $x$ and $\frac{2}{y}$ .

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

Reorder terms.

$\frac{\frac{4 x^{2} - y^{2}}{x y} x}{\frac{2 x}{y}} - \frac{\frac{2}{y}}{\frac{2 x}{y}}$

Step 4

Combine the numerators over the common denominator.

$\frac{\frac{4 x^{2} - y^{2}}{x y} x - \frac{2}{y}}{\frac{2 x}{y}}$

Step 5

Rewrite $\frac{\frac{4 x^{2} - y^{2}}{x y} x - \frac{2}{y}}{\frac{2 x}{y}}$ in a factored form.

Cancel the common factor of $x$ .

Factor $x$ out of $x y$ .

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

Cancel the common factor.

$\frac{\frac{4 x^{2} - y^{2}}{x y} x - \frac{2}{y}}{\frac{2 x}{y}}$

Rewrite the expression.

$\frac{\frac{4 x^{2} - y^{2}}{y} - \frac{2}{y}}{\frac{2 x}{y}}$

Simplify the numerator.

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

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

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

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

Combine the numerators over the common denominator.

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

Rewrite $\frac{(2 x + y) (2 x - y) - 2}{y}$ in a factored form.

Expand $(2 x + y) (2 x - y)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $2$ by $2$ .

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

Rewrite using the commutative property of multiplication.

$\frac{\frac{4 x^{2} + 2 \cdot - 1 x y + y (2 x) + y (- y) - 2}{y}}{\frac{2 x}{y}}$

Multiply $2$ by $- 1$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

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

Move $y$ .

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

Multiply $y$ by $y$ .

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

Add $- 2 x y$ and $2 y x$ .

Move $y$ .

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

Add $- 2 x y$ and $2 x y$ .

$\frac{\frac{4 x^{2} + 0 - y^{2} - 2}{y}}{\frac{2 x}{y}}$

Add $4 x^{2}$ and $0$ .

$\frac{\frac{4 x^{2} - y^{2} - 2}{y}}{\frac{2 x}{y}}$

Factor over the Complex Numbers ((4x^2-y^2)/(xy))÷(2/y)-1/x

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