E= \(\frac{3x^2-8x+6}{x^2-2x+1}\)= \(\frac{3x^2-8x+6}{\left(x-1\right)^2}\)
Đặt t= x-1 => x= t+1
E= \(\frac{3\left(t+1\right)^2-8\left(t+1\right)+6}{t^2}\)= \(\frac{3\left(t^2+2t+1\right)-8t-8+6}{t^2}\)=\(\frac{3t^2+6t+3-8t-8+6}{t^2}\)=\(\frac{3t^2-2t+1}{t^2}\)= \(\frac{3t^2}{t^2}\)\(-\frac{2t}{t^2}\)\(\frac{1}{t^2}\)= 3-\(\frac{2}{t}\)+\(\frac{1}{t^2}\)
Đặt \(\frac{1}{t}\)= b \(\Rightarrow\)E= b2-2b+3= b2-2b+1+2= (b-1)2+2\(\ge\)2
Min E= 2 khi b-1= 0 \(\Rightarrow\)b= 1\(\Rightarrow\)\(\frac{1}{t}\)= 1\(\Rightarrow\)\(\frac{1}{x-1}\)= 1 \(\Leftrightarrow\)x-1= 1 \(\Rightarrow\)x=1