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

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

Step 1

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

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

Apply the distributive property.

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

Multiply $a$ by $a$ .

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

Apply the distributive property.

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

Subtract $a^{2}$ from $a^{2}$ .

$b + \frac{a b + 0 - b^{2}}{a + b}$

Add $a b$ and $0$ .

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

Factor $b$ out of $a b - b^{2}$ .

Factor $b$ out of $a b$ .

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

Factor $b$ out of $- b^{2}$ .

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

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

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

Step 4

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

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

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

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

Add $a$ and $a$ .

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

Subtract $b$ from $b$ .

$\frac{b (2 a + 0)}{a + b}$

Add $2 a$ and $0$ .

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

Step 7

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

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