Solve (c+d/c-2c/c-d)d-c/c^2+d^2

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

Evaluate

$$\frac{1}{c}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of c and c-d is c\left(c-d\right). Multiply \frac{c+d}{c} times \frac{c-d}{c-d}. Multiply \frac{2c}{c-d} times \frac{c}{c}.

$$\left(\frac{\left(c+d\right)\left(c-d\right)}{c\left(c-d\right)}-\frac{2cc}{c\left(c-d\right)}\right)\times \frac{d-c}{c^{2}+d^{2}}$$

Since \frac{\left(c+d\right)\left(c-d\right)}{c\left(c-d\right)} and \frac{2cc}{c\left(c-d\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(c+d\right)\left(c-d\right)-2cc}{c\left(c-d\right)}\times \frac{d-c}{c^{2}+d^{2}}$$

Do the multiplications in \left(c+d\right)\left(c-d\right)-2cc.

$$\frac{c^{2}-cd+dc-d^{2}-2c^{2}}{c\left(c-d\right)}\times \frac{d-c}{c^{2}+d^{2}}$$

Combine like terms in c^{2}-cd+dc-d^{2}-2c^{2}.

$$\frac{-c^{2}-d^{2}}{c\left(c-d\right)}\times \frac{d-c}{c^{2}+d^{2}}$$

Multiply \frac{-c^{2}-d^{2}}{c\left(c-d\right)} times \frac{d-c}{c^{2}+d^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-c^{2}-d^{2}\right)\left(d-c\right)}{c\left(c-d\right)\left(c^{2}+d^{2}\right)}$$

Extract the negative sign in -c^{2}-d^{2}. Extract the negative sign in d-c.

$$\frac{-\left(-1\right)\left(c-d\right)\left(c^{2}+d^{2}\right)}{c\left(c-d\right)\left(c^{2}+d^{2}\right)}$$

Cancel out \left(c-d\right)\left(c^{2}+d^{2}\right) in both numerator and denominator.

$$\frac{-\left(-1\right)}{c}$$

Multiply -1 and -1 to get 1.

$$\frac{1}{c}$$

Expand

$$\frac{1}{c}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of c and c-d is c\left(c-d\right). Multiply \frac{c+d}{c} times \frac{c-d}{c-d}. Multiply \frac{2c}{c-d} times \frac{c}{c}.

$$\left(\frac{\left(c+d\right)\left(c-d\right)}{c\left(c-d\right)}-\frac{2cc}{c\left(c-d\right)}\right)\times \frac{d-c}{c^{2}+d^{2}}$$

Since \frac{\left(c+d\right)\left(c-d\right)}{c\left(c-d\right)} and \frac{2cc}{c\left(c-d\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\left(c+d\right)\left(c-d\right)-2cc}{c\left(c-d\right)}\times \frac{d-c}{c^{2}+d^{2}}$$

Do the multiplications in \left(c+d\right)\left(c-d\right)-2cc.

$$\frac{c^{2}-cd+dc-d^{2}-2c^{2}}{c\left(c-d\right)}\times \frac{d-c}{c^{2}+d^{2}}$$

Combine like terms in c^{2}-cd+dc-d^{2}-2c^{2}.

$$\frac{-c^{2}-d^{2}}{c\left(c-d\right)}\times \frac{d-c}{c^{2}+d^{2}}$$

Multiply \frac{-c^{2}-d^{2}}{c\left(c-d\right)} times \frac{d-c}{c^{2}+d^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(-c^{2}-d^{2}\right)\left(d-c\right)}{c\left(c-d\right)\left(c^{2}+d^{2}\right)}$$

Extract the negative sign in -c^{2}-d^{2}. Extract the negative sign in d-c.

$$\frac{-\left(-1\right)\left(c-d\right)\left(c^{2}+d^{2}\right)}{c\left(c-d\right)\left(c^{2}+d^{2}\right)}$$

Cancel out \left(c-d\right)\left(c^{2}+d^{2}\right) in both numerator and denominator.

$$\frac{-\left(-1\right)}{c}$$

Multiply -1 and -1 to get 1.

$$\frac{1}{c}$$