Simplify (b-c)/b+b/(b+c)

$\frac{b - c}{b} + \frac{b}{b + c}$

Step 1

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

$\frac{b - c}{b} \cdot \frac{b + c}{b + c} + \frac{b}{b + c}$

Step 2

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

$\frac{b - c}{b} \cdot \frac{b + c}{b + c} + \frac{b}{b + c} \cdot \frac{b}{b}$

Step 3

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

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

$\frac{(b - c) (b + c)}{b (b + c)} + \frac{b}{b + c} \cdot \frac{b}{b}$

Multiply $\frac{b}{b + c}$ by $\frac{b}{b}$ .

$\frac{(b - c) (b + c)}{b (b + c)} + \frac{b \cdot b}{(b + c) b}$

Reorder the factors of $(b + c) b$ .

$\frac{(b - c) (b + c)}{b (b + c)} + \frac{b \cdot b}{b (b + c)}$

Step 4

Combine the numerators over the common denominator.

$\frac{(b - c) (b + c) + b \cdot b}{b (b + c)}$

Step 5

Simplify the numerator.

Expand $(b - c) (b + c)$ using the FOIL Method.

Apply the distributive property.

$\frac{b (b + c) - c (b + c) + b \cdot b}{b (b + c)}$

Apply the distributive property.

$\frac{b \cdot b + b c - c (b + c) + b \cdot b}{b (b + c)}$

Apply the distributive property.

$\frac{b \cdot b + b c - c b - c \cdot c + b \cdot b}{b (b + c)}$

Combine the opposite terms in $b \cdot b + b c - c b - c \cdot c$ .

Reorder the factors in the terms $b c$ and $- c b$ .

$\frac{b \cdot b + b c - b c - c \cdot c + b \cdot b}{b (b + c)}$

Subtract $b c$ from $b c$ .

$\frac{b \cdot b + 0 - c \cdot c + b \cdot b}{b (b + c)}$

Add $b \cdot b$ and $0$ .

$\frac{b \cdot b - c \cdot c + b \cdot b}{b (b + c)}$

Simplify each term.

Multiply $b$ by $b$ .

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

Multiply $c$ by $c$ by adding the exponents.

Move $c$ .

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

Multiply $c$ by $c$ .

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

Multiply $b$ by $b$ .

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

Add $b^{2}$ and $b^{2}$ .

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

Simplify (b-c)/b+b/(b+c)

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