Solve x^2-10x+2x/2x(x/x^2+2x-x/x-5)^2+5/5+x

$$\frac { x ^ { 2 } - 10 x + 2 x } { 2 x } ( \frac { x } { x ^ { 2 } + 2 x } - \frac { x } { x - 5 } ) ^ { 2 } + \frac { 5 } { 5 + x }$$

Evaluate

$$\frac{x\left(x^{5}-x^{4}-25x^{3}-163x^{2}-555x+125\right)}{2\left(x+5\right)\left(\left(x-5\right)\left(x+2\right)\right)^{2}}$$

Short Solution Steps

Combine -10x and 2x to get -8x.

$$\frac{x^{2}-8x}{2x}\left(\frac{x}{x^{2}+2x}-\frac{x}{x-5}\right)^{2}+\frac{5}{5+x}$$

Factor the expressions that are not already factored in \frac{x^{2}-8x}{2x}.

$$\frac{x\left(x-8\right)}{2x}\left(\frac{x}{x^{2}+2x}-\frac{x}{x-5}\right)^{2}+\frac{5}{5+x}$$

Cancel out x in both numerator and denominator.

$$\frac{x-8}{2}\left(\frac{x}{x^{2}+2x}-\frac{x}{x-5}\right)^{2}+\frac{5}{5+x}$$

Factor the expressions that are not already factored in \frac{x}{x^{2}+2x}.

$$\frac{x-8}{2}\left(\frac{x}{x\left(x+2\right)}-\frac{x}{x-5}\right)^{2}+\frac{5}{5+x}$$

Cancel out x in both numerator and denominator.

$$\frac{x-8}{2}\left(\frac{1}{x+2}-\frac{x}{x-5}\right)^{2}+\frac{5}{5+x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x-5 is \left(x-5\right)\left(x+2\right). Multiply \frac{1}{x+2} times \frac{x-5}{x-5}. Multiply \frac{x}{x-5} times \frac{x+2}{x+2}.

$$\frac{x-8}{2}\left(\frac{x-5}{\left(x-5\right)\left(x+2\right)}-\frac{x\left(x+2\right)}{\left(x-5\right)\left(x+2\right)}\right)^{2}+\frac{5}{5+x}$$

Since \frac{x-5}{\left(x-5\right)\left(x+2\right)} and \frac{x\left(x+2\right)}{\left(x-5\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x-8}{2}\times \left(\frac{x-5-x\left(x+2\right)}{\left(x-5\right)\left(x+2\right)}\right)^{2}+\frac{5}{5+x}$$

Do the multiplications in x-5-x\left(x+2\right).

$$\frac{x-8}{2}\times \left(\frac{x-5-x^{2}-2x}{\left(x-5\right)\left(x+2\right)}\right)^{2}+\frac{5}{5+x}$$

Combine like terms in x-5-x^{2}-2x.

$$\frac{x-8}{2}\times \left(\frac{-x-5-x^{2}}{\left(x-5\right)\left(x+2\right)}\right)^{2}+\frac{5}{5+x}$$

To raise \frac{-x-5-x^{2}}{\left(x-5\right)\left(x+2\right)} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{x-8}{2}\times \frac{\left(-x-5-x^{2}\right)^{2}}{\left(\left(x-5\right)\left(x+2\right)\right)^{2}}+\frac{5}{5+x}$$

Multiply \frac{x-8}{2} times \frac{\left(-x-5-x^{2}\right)^{2}}{\left(\left(x-5\right)\left(x+2\right)\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x-8\right)\left(-x-5-x^{2}\right)^{2}}{2\left(\left(x-5\right)\left(x+2\right)\right)^{2}}+\frac{5}{5+x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(\left(x-5\right)\left(x+2\right)\right)^{2} and 5+x is 2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}. Multiply \frac{\left(x-8\right)\left(-x-5-x^{2}\right)^{2}}{2\left(\left(x-5\right)\left(x+2\right)\right)^{2}} times \frac{x+5}{x+5}. Multiply \frac{5}{5+x} times \frac{2\left(x-5\right)^{2}\left(x+2\right)^{2}}{2\left(x-5\right)^{2}\left(x+2\right)^{2}}.

$$\frac{\left(x-8\right)\left(-x-5-x^{2}\right)^{2}\left(x+5\right)}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}}+\frac{5\times 2\left(x-5\right)^{2}\left(x+2\right)^{2}}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}}$$

Since \frac{\left(x-8\right)\left(-x-5-x^{2}\right)^{2}\left(x+5\right)}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}} and \frac{5\times 2\left(x-5\right)^{2}\left(x+2\right)^{2}}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-8\right)\left(-x-5-x^{2}\right)^{2}\left(x+5\right)+5\times 2\left(x-5\right)^{2}\left(x+2\right)^{2}}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}}$$

Do the multiplications in \left(x-8\right)\left(-x-5-x^{2}\right)^{2}\left(x+5\right)+5\times 2\left(x-5\right)^{2}\left(x+2\right)^{2}.

$$\frac{21x^{4}+65x^{3}+75x^{2}+7x^{5}+125x+x^{6}-168x^{3}-520x^{2}-600x-56x^{4}-1000-8x^{5}+10x^{4}+40x^{3}+40x^{2}-100x^{3}-400x^{2}-400x+250x^{2}+1000x+1000}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}}$$

Combine like terms in 21x^{4}+65x^{3}+75x^{2}+7x^{5}+125x+x^{6}-168x^{3}-520x^{2}-600x-56x^{4}-1000-8x^{5}+10x^{4}+40x^{3}+40x^{2}-100x^{3}-400x^{2}-400x+250x^{2}+1000x+1000.

$$\frac{-25x^{4}-163x^{3}-555x^{2}-x^{5}+125x+x^{6}}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}}$$

Expand 2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}.

$$\frac{-25x^{4}-163x^{3}-555x^{2}-x^{5}+125x+x^{6}}{2x^{5}-2x^{4}-82x^{3}+10x^{2}+800x+1000}$$

Expand

$$\frac{x^{6}-x^{5}-25x^{4}-163x^{3}-555x^{2}+125x}{2\left(x+5\right)\left(\left(x-5\right)\left(x+2\right)\right)^{2}}$$

Short Solution Steps

Combine -10x and 2x to get -8x.

$$\frac{x^{2}-8x}{2x}\left(\frac{x}{x^{2}+2x}-\frac{x}{x-5}\right)^{2}+\frac{5}{5+x}$$

Factor the expressions that are not already factored in \frac{x^{2}-8x}{2x}.

$$\frac{x\left(x-8\right)}{2x}\left(\frac{x}{x^{2}+2x}-\frac{x}{x-5}\right)^{2}+\frac{5}{5+x}$$

Cancel out x in both numerator and denominator.

$$\frac{x-8}{2}\left(\frac{x}{x^{2}+2x}-\frac{x}{x-5}\right)^{2}+\frac{5}{5+x}$$

Factor the expressions that are not already factored in \frac{x}{x^{2}+2x}.

$$\frac{x-8}{2}\left(\frac{x}{x\left(x+2\right)}-\frac{x}{x-5}\right)^{2}+\frac{5}{5+x}$$

Cancel out x in both numerator and denominator.

$$\frac{x-8}{2}\left(\frac{1}{x+2}-\frac{x}{x-5}\right)^{2}+\frac{5}{5+x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of x+2 and x-5 is \left(x-5\right)\left(x+2\right). Multiply \frac{1}{x+2} times \frac{x-5}{x-5}. Multiply \frac{x}{x-5} times \frac{x+2}{x+2}.

$$\frac{x-8}{2}\left(\frac{x-5}{\left(x-5\right)\left(x+2\right)}-\frac{x\left(x+2\right)}{\left(x-5\right)\left(x+2\right)}\right)^{2}+\frac{5}{5+x}$$

Since \frac{x-5}{\left(x-5\right)\left(x+2\right)} and \frac{x\left(x+2\right)}{\left(x-5\right)\left(x+2\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{x-8}{2}\times \left(\frac{x-5-x\left(x+2\right)}{\left(x-5\right)\left(x+2\right)}\right)^{2}+\frac{5}{5+x}$$

Do the multiplications in x-5-x\left(x+2\right).

$$\frac{x-8}{2}\times \left(\frac{x-5-x^{2}-2x}{\left(x-5\right)\left(x+2\right)}\right)^{2}+\frac{5}{5+x}$$

Combine like terms in x-5-x^{2}-2x.

$$\frac{x-8}{2}\times \left(\frac{-x-5-x^{2}}{\left(x-5\right)\left(x+2\right)}\right)^{2}+\frac{5}{5+x}$$

To raise \frac{-x-5-x^{2}}{\left(x-5\right)\left(x+2\right)} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{x-8}{2}\times \frac{\left(-x-5-x^{2}\right)^{2}}{\left(\left(x-5\right)\left(x+2\right)\right)^{2}}+\frac{5}{5+x}$$

Multiply \frac{x-8}{2} times \frac{\left(-x-5-x^{2}\right)^{2}}{\left(\left(x-5\right)\left(x+2\right)\right)^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{\left(x-8\right)\left(-x-5-x^{2}\right)^{2}}{2\left(\left(x-5\right)\left(x+2\right)\right)^{2}}+\frac{5}{5+x}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2\left(\left(x-5\right)\left(x+2\right)\right)^{2} and 5+x is 2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}. Multiply \frac{\left(x-8\right)\left(-x-5-x^{2}\right)^{2}}{2\left(\left(x-5\right)\left(x+2\right)\right)^{2}} times \frac{x+5}{x+5}. Multiply \frac{5}{5+x} times \frac{2\left(x-5\right)^{2}\left(x+2\right)^{2}}{2\left(x-5\right)^{2}\left(x+2\right)^{2}}.

$$\frac{\left(x-8\right)\left(-x-5-x^{2}\right)^{2}\left(x+5\right)}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}}+\frac{5\times 2\left(x-5\right)^{2}\left(x+2\right)^{2}}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}}$$

Since \frac{\left(x-8\right)\left(-x-5-x^{2}\right)^{2}\left(x+5\right)}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}} and \frac{5\times 2\left(x-5\right)^{2}\left(x+2\right)^{2}}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\left(x-8\right)\left(-x-5-x^{2}\right)^{2}\left(x+5\right)+5\times 2\left(x-5\right)^{2}\left(x+2\right)^{2}}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}}$$

Do the multiplications in \left(x-8\right)\left(-x-5-x^{2}\right)^{2}\left(x+5\right)+5\times 2\left(x-5\right)^{2}\left(x+2\right)^{2}.

$$\frac{21x^{4}+65x^{3}+75x^{2}+7x^{5}+125x+x^{6}-168x^{3}-520x^{2}-600x-56x^{4}-1000-8x^{5}+10x^{4}+40x^{3}+40x^{2}-100x^{3}-400x^{2}-400x+250x^{2}+1000x+1000}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}}$$

Combine like terms in 21x^{4}+65x^{3}+75x^{2}+7x^{5}+125x+x^{6}-168x^{3}-520x^{2}-600x-56x^{4}-1000-8x^{5}+10x^{4}+40x^{3}+40x^{2}-100x^{3}-400x^{2}-400x+250x^{2}+1000x+1000.

$$\frac{-25x^{4}-163x^{3}-555x^{2}-x^{5}+125x+x^{6}}{2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}}$$

Expand 2\left(x+5\right)\left(x-5\right)^{2}\left(x+2\right)^{2}.

$$\frac{-25x^{4}-163x^{3}-555x^{2}-x^{5}+125x+x^{6}}{2x^{5}-2x^{4}-82x^{3}+10x^{2}+800x+1000}$$