Với \(m\ne0\) có: \(\Delta'=\left(m-2\right)^2-m\left(m-3\right)=4-m\ge0\Rightarrow m\le4\)
Theo Viet: \(\left\{{}\begin{matrix}x_1+x_2=\frac{-2\left(m-2\right)}{m}=-2\left(1-\frac{2}{m}\right)\\x_1x_2=\frac{m-3}{m}=1-\frac{3}{m}\end{matrix}\right.\)
\(A=x_1^2+x_2^2=\left(x_1+x_2\right)^2-2x_1x_2\)
\(=4\left(1-\frac{2}{m}\right)^2-2\left(1-\frac{3}{m}\right)=4\left(\frac{4}{m^2}-\frac{4}{m}+1\right)-2+\frac{6}{m}\)
\(=\frac{16}{m^2}-\frac{10}{m}+2=16\left(\frac{1}{m}-\frac{5}{16}\right)^2+\frac{7}{16}\ge\frac{7}{16}\)
\(A_{min}=\frac{7}{16}\) khi \(\frac{1}{m}=\frac{5}{16}\Leftrightarrow m=\frac{16}{5}< 4\left(t/m\right)\)
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