Question

Tìm m để phương trình có 4 nghiệm:

\(x^4-6x^3+5x^2+\left(2m+12\right)x-m^2-3m=0\)

Answer 1

\(\Leftrightarrow x^4-2x^3-\left(m+3\right)x^2-4x^3+8x^2+4\left(m+3\right)x+mx^2-2mx-m^2-3m=0\)

\(\Leftrightarrow x^2\left(x^2-2x-m-3\right)-4x\left(x^2-2x-m-3\right)+m\left(x^2-2x-m-3\right)=0\)

\(\Leftrightarrow\left(x^2-4x+m\right)\left(x^2-2x-m-3\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2-4x+m=0\\x^2-2x-m-3=0\end{matrix}\right.\)

Pt có 4 nghiệm khi: \(\left\{{}\begin{matrix}\Delta'_1=4-m\ge0\\\Delta'_2=1+m+3\ge0\end{matrix}\right.\)

\(\Leftrightarrow-4\le m\le4\)