Simplify (2ab)/(a^2-b^2)-b/(a-b)+2

$\frac{2 a b}{a^{2} - b^{2}} - \frac{b}{a - b} + 2$

Step 1

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

$\frac{2 a b}{(a + b) (a - b)} - \frac{b}{a - b} + 2$

Step 2

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

$\frac{2 a b}{(a + b) (a - b)} - \frac{b}{a - b} \cdot \frac{a + b}{a + b} + 2$

Step 3

Write each expression with a common denominator of $(a + b) (a - b)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{b}{a - b}$ by $\frac{a + b}{a + b}$ .

$\frac{2 a b}{(a + b) (a - b)} - \frac{b (a + b)}{(a - b) (a + b)} + 2$

Reorder the factors of $(a - b) (a + b)$ .

$\frac{2 a b}{(a + b) (a - b)} - \frac{b (a + b)}{(a + b) (a - b)} + 2$

Step 4

Combine the numerators over the common denominator.

$\frac{2 a b - b (a + b)}{(a + b) (a - b)} + 2$

Step 5

Simplify each term.

Simplify the numerator.

Factor $b$ out of $2 a b - b (a + b)$ .

Factor $b$ out of $2 a b$ .

$\frac{b (2 a) - b (a + b)}{(a + b) (a - b)} + 2$

Factor $b$ out of $- b (a + b)$ .

$\frac{b (2 a) + b (- (a + b))}{(a + b) (a - b)} + 2$

Factor $b$ out of $b (2 a) + b (- (a + b))$ .

$\frac{b (2 a - (a + b))}{(a + b) (a - b)} + 2$

Apply the distributive property.

$\frac{b (2 a - a - b)}{(a + b) (a - b)} + 2$

Subtract $a$ from $2 a$ .

$\frac{b (a - b)}{(a + b) (a - b)} + 2$

Cancel the common factor of $a - b$ .

Cancel the common factor.

$\frac{b (a - b)}{(a + b) (a - b)} + 2$

Rewrite the expression.

$\frac{b}{a + b} + 2$

Step 6

To write $2$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

$\frac{b}{a + b} + \frac{2 (a + b)}{a + b}$

Step 7

Combine the numerators over the common denominator.

$\frac{b + 2 (a + b)}{a + b}$

Step 8

Simplify the numerator.

Apply the distributive property.

$\frac{b + 2 a + 2 b}{a + b}$

Add $b$ and $2 b$ .

$\frac{2 a + 3 b}{a + b}$

Simplify (2ab)/(a^2-b^2)-b/(a-b)+2

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