Question

Tìm m để phương trình 2x³ - 7x² +2( 5-m)x + 3(m-2) =0 có ba nghiệm phân biệt

Answer 1

\(\Leftrightarrow2x^3-7x^2+10x-6-m\left(2x-3\right)=0\)

\(\Leftrightarrow\left(2x-3\right)\left(x^2-2x+2\right)-m\left(2x-3\right)=0\)

\(\Leftrightarrow\left(2x-3\right)\left(x^2-2x-m+2\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x=\frac{3}{2}\\x^2-2x-m+2=0\left(1\right)\end{matrix}\right.\)

Để pt đã cho có 3 nghiệm pb \(\Leftrightarrow\left(1\right)\) có 2 nghiệm pb khác \(\frac{3}{2}\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(\frac{3}{2}\right)^2-2\left(\frac{3}{2}\right)-m+2\ne0\\\Delta'=1-\left(-m+2\right)>0\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}m\ne\frac{5}{4}\\m>-1\end{matrix}\right.\)