\(\dfrac{1}{\left(x+2\right)\left(x+3\right)}+\dfrac{1}{\left(x+3\right)\left(x+4\right)}+.....+\dfrac{1}{\left(x+99\right)\left(x+100\right)}=5\)
\(\Leftrightarrow\dfrac{1}{x+2}-\dfrac{1}{x+3}+\dfrac{1}{x+3}-\dfrac{1}{x+4}+.....+\dfrac{1}{x+99}-\dfrac{1}{x+100}=5\)
\(\Leftrightarrow\dfrac{1}{x+2}-\dfrac{1}{x+100}=5\)
\(\Leftrightarrow\dfrac{x+100-x-2}{\left(x+2\right)\left(x+100\right)}=5\)
\(\Leftrightarrow98=5\left(x+2\right)\left(x+100\right)\)
\(\Leftrightarrow98=5\left(x^2+102x+200\right)\)
\(\Leftrightarrow5x^2+510x+1000-98=0\)
\(\Leftrightarrow5x^2+510x+902=0\)
\(\Leftrightarrow5(x^2+102x+\dfrac{902}{5})=0\)
\(\Leftrightarrow(x^2+2.x.51+51^2-2601+\dfrac{902}{5})=0\)
\(\Leftrightarrow\left[\left(x+51\right)^2-\dfrac{12103}{5}\right]=0\)
\(\Leftrightarrow\left(x+51\right)^2=\dfrac{12103}{5}\)
\(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{12103}{5}-51\\x=-\dfrac{12103}{5}-51\end{matrix}\right.\)
