Factor over the Complex Numbers ((25x^2-y^2)/(xy))÷(5/y)-1/x

$\frac{25 x^{2} - y^{2}}{x y} \div \frac{5}{y} - \frac{1}{x}$

Step 1

To write $\frac{25 x^{2} - y^{2}}{x y} \div \frac{5}{y}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{25 x^{2} - y^{2}}{x y} \div \frac{5}{y} \cdot \frac{x}{x} - \frac{1}{x}$

Step 2

To write $- \frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{\frac{5}{y}}{\frac{5}{y}}$ .

$\frac{25 x^{2} - y^{2}}{x y} \div \frac{5}{y} \cdot \frac{x}{x} - \frac{1}{x} \cdot \frac{\frac{5}{y}}{\frac{5}{y}}$

Step 3

Write each expression with a common denominator of $\frac{5}{y} x$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{25 x^{2} - y^{2}}{x y} \div \frac{5}{y}$ by $\frac{x}{x}$ .

$\frac{\frac{25 x^{2} - y^{2}}{x y} x}{\frac{5}{y} x} - \frac{1}{x} \cdot \frac{\frac{5}{y}}{\frac{5}{y}}$

Combine $\frac{5}{y}$ and $x$ .

$\frac{\frac{25 x^{2} - y^{2}}{x y} x}{\frac{5 x}{y}} - \frac{1}{x} \cdot \frac{\frac{5}{y}}{\frac{5}{y}}$

Multiply $\frac{1}{x}$ by $\frac{\frac{5}{y}}{\frac{5}{y}}$ .

$\frac{\frac{25 x^{2} - y^{2}}{x y} x}{\frac{5 x}{y}} - \frac{\frac{5}{y}}{x \frac{5}{y}}$

Combine $x$ and $\frac{5}{y}$ .

$\frac{\frac{25 x^{2} - y^{2}}{x y} x}{\frac{5 x}{y}} - \frac{\frac{5}{y}}{\frac{x \cdot 5}{y}}$

Reorder terms.

$\frac{\frac{25 x^{2} - y^{2}}{x y} x}{\frac{5 x}{y}} - \frac{\frac{5}{y}}{\frac{5 x}{y}}$

Step 4

Combine the numerators over the common denominator.

$\frac{\frac{25 x^{2} - y^{2}}{x y} x - \frac{5}{y}}{\frac{5 x}{y}}$

Step 5

Rewrite $\frac{\frac{25 x^{2} - y^{2}}{x y} x - \frac{5}{y}}{\frac{5 x}{y}}$ in a factored form.

Cancel the common factor of $x$ .

Factor $x$ out of $x y$ .

$\frac{\frac{25 x^{2} - y^{2}}{x (y)} x - \frac{5}{y}}{\frac{5 x}{y}}$

Cancel the common factor.

$\frac{\frac{25 x^{2} - y^{2}}{x y} x - \frac{5}{y}}{\frac{5 x}{y}}$

Rewrite the expression.

$\frac{\frac{25 x^{2} - y^{2}}{y} - \frac{5}{y}}{\frac{5 x}{y}}$

Simplify the numerator.

Rewrite $25 x^{2}$ as ${(5 x)}^{2}$ .

$\frac{\frac{{(5 x)}^{2} - y^{2}}{y} - \frac{5}{y}}{\frac{5 x}{y}}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - i^{2} = (a + i) (a - i)$ where $a = 5 x$ and $i = y$ .

$\frac{\frac{(5 x + y) (5 x - y)}{y} - \frac{5}{y}}{\frac{5 x}{y}}$

Combine the numerators over the common denominator.

$\frac{\frac{(5 x + y) (5 x - y) - 5}{y}}{\frac{5 x}{y}}$

Rewrite $\frac{(5 x + y) (5 x - y) - 5}{y}$ in a factored form.

Expand $(5 x + y) (5 x - y)$ using the FOIL Method.

Apply the distributive property.

$\frac{\frac{5 x (5 x - y) + y (5 x - y) - 5}{y}}{\frac{5 x}{y}}$

Apply the distributive property.

$\frac{\frac{5 x (5 x) + 5 x (- y) + y (5 x - y) - 5}{y}}{\frac{5 x}{y}}$

Apply the distributive property.

$\frac{\frac{5 x (5 x) + 5 x (- y) + y (5 x) + y (- y) - 5}{y}}{\frac{5 x}{y}}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$\frac{\frac{5 \cdot 5 x \cdot x + 5 x (- y) + y (5 x) + y (- y) - 5}{y}}{\frac{5 x}{y}}$

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

Move $x$ .

$\frac{\frac{5 \cdot 5 (x \cdot x) + 5 x (- y) + y (5 x) + y (- y) - 5}{y}}{\frac{5 x}{y}}$

Multiply $x$ by $x$ .

$\frac{\frac{5 \cdot 5 x^{2} + 5 x (- y) + y (5 x) + y (- y) - 5}{y}}{\frac{5 x}{y}}$

Multiply $5$ by $5$ .

$\frac{\frac{25 x^{2} + 5 x (- y) + y (5 x) + y (- y) - 5}{y}}{\frac{5 x}{y}}$

Rewrite using the commutative property of multiplication.

$\frac{\frac{25 x^{2} + 5 \cdot - 1 x y + y (5 x) + y (- y) - 5}{y}}{\frac{5 x}{y}}$

Multiply $5$ by $- 1$ .

$\frac{\frac{25 x^{2} - 5 x y + y (5 x) + y (- y) - 5}{y}}{\frac{5 x}{y}}$

Rewrite using the commutative property of multiplication.

$\frac{\frac{25 x^{2} - 5 x y + 5 y x + y (- y) - 5}{y}}{\frac{5 x}{y}}$

Rewrite using the commutative property of multiplication.

$\frac{\frac{25 x^{2} - 5 x y + 5 y x - y \cdot y - 5}{y}}{\frac{5 x}{y}}$

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

Move $y$ .

$\frac{\frac{25 x^{2} - 5 x y + 5 y x - (y \cdot y) - 5}{y}}{\frac{5 x}{y}}$

Multiply $y$ by $y$ .

$\frac{\frac{25 x^{2} - 5 x y + 5 y x - y^{2} - 5}{y}}{\frac{5 x}{y}}$

Add $- 5 x y$ and $5 y x$ .

Move $y$ .

$\frac{\frac{25 x^{2} - 5 x y + 5 x y - y^{2} - 5}{y}}{\frac{5 x}{y}}$

Add $- 5 x y$ and $5 x y$ .

$\frac{\frac{25 x^{2} + 0 - y^{2} - 5}{y}}{\frac{5 x}{y}}$

Add $25 x^{2}$ and $0$ .

$\frac{\frac{25 x^{2} - y^{2} - 5}{y}}{\frac{5 x}{y}}$

Factor over the Complex Numbers ((25x^2-y^2)/(xy))÷(5/y)-1/x

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