\(\dfrac{\left(2009-x\right)^2+\left(2009-x\right)\left(x-2010\right)+\left(x-2010\right)^2}{\left(2009-x\right)^2-\left(2009-x\right)\left(x-2010\right)+\left(x-2010\right)^2}=\dfrac{19}{49}\left(1\right)\)

\(Đkxđ:x\ne2009;x\ne2010\)

Đặt \(t=x-2010\left(t\ne0\right)\)

\(\Rightarrow2009-x=-\left(t+1\right)\)

\(\left(1\right)\Leftrightarrow\dfrac{\left(t+1\right)^2-\left(t+1\right)t+t^2}{\left(t+1\right)^2+\left(t+1\right)t+t^2}=\dfrac{19}{49}\)

\(\Leftrightarrow\dfrac{t^2+2t+1-t^2-t+t^2}{t^2+2t+1+t^2+t+t^2}=\dfrac{19}{49}\)

\(\Leftrightarrow\dfrac{t^2+t+1}{3t^2+3t+1}=\dfrac{19}{49}\)

\(\Leftrightarrow49t^2+49t+49=57t^2+57t+19\)

\(\Leftrightarrow8t^2+8t-30=0\)

\(\Leftrightarrow4t^2+4t-15=0\)

\(\Leftrightarrow\left(4t^2+4t+1\right)-16=0\)

\(\Leftrightarrow\left(2t+1\right)^2=16=4^2\)

\(\Leftrightarrow\left[{}\begin{matrix}2t+1=4\\2t+1=-4\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}t=\dfrac{3}{2}\\t=-\dfrac{5}{2}\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x-2010=\dfrac{3}{2}\\x-2010=-\dfrac{5}{2}\end{matrix}\right.\) \(\Leftrightarrow\left[{}\begin{matrix}x=\dfrac{4023}{2}\\x=\dfrac{4015}{2}\end{matrix}\right.\)

\(\Leftrightarrow\dfrac{x+1}{2010}+1+\dfrac{x+2}{2009}+1+...+\dfrac{x+2009}{2}+1+\dfrac{x+2010}{1}+1=0\)

=>x+2011=0

hay x=-2011

dat a =2009-x

b=x-2010

ta co : a^2+ab+b^2/a^2-ab+b^2 =19/49

<=>49a^2+49ab+49b^2=19a^2-19a+19b^2

<=>30a^2+68a+30b^2=0

<=>15a^2+34ab+15b^2=0

<=>15a^2+9ab+25ab+15b^2=0

<=>3a(5a+3b)+5b(5a+3b)=0

<=>(5a+3b)(3a+5b)=0

<=>5a+3b=0 hoac 3a+5b=0

vs 5a +3b=0 <=>5(2009-x)+3(x-2010)=0=>x=......