Simplify (4xy)/(x^2-y^2)+(x-y)/(x+y)

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

Step 1

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 $b = y$ .

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

Step 2

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

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

Step 3

Simplify terms.

Multiply $\frac{x - y}{x + y}$ by $\frac{x - y}{x - y}$ .

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

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Rewrite using the commutative property of multiplication.

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

Rewrite using the commutative property of multiplication.

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

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

Move $y$ .

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

Multiply $y$ by $y$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $y^{2}$ by $1$ .

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

Subtract $y x$ from $- x y$ .

Move $y$ .

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

Subtract $x y$ from $- x y$ .

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

Subtract $2 x y$ from $4 x y$ .

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

Factor using the perfect square rule.

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x y = 2 \cdot x \cdot y$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = y$ .

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

Step 5

Cancel the common factor of ${(x + y)}^{2}$ and $x + y$ .

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

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

Cancel the common factors.

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + y}{x - y}$

Simplify (4xy)/(x^2-y^2)+(x-y)/(x+y)

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