Question

[Nguyễn Ngọc Gia Hân]

Giải phương trình:

a.2x⁴-21x³+74x²-105x+50=0

b.x⁴-x³-10x²+2x+4=0

Answer 1

[Y]

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

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

\(\Leftrightarrow\left[{}\begin{matrix}x^2-3x-2=0\\x^2+2x-2=0\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}\left(x-\frac{3}{2}\right)^2=\frac{17}{4}\\\left(x+1\right)^2=3\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x-\frac{3}{2}=\frac{\sqrt{17}}{2}\\x-\frac{3}{2}=-\frac{\sqrt{17}}{2}\\x+1=\sqrt{3}\\x+1=-\sqrt{3}\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x=\frac{3+\sqrt{17}}{2}\\x=\frac{3-\sqrt{17}}{2}\\x=\sqrt{3}-1\\x=-1-\sqrt{3}\end{matrix}\right.\) ( TM )

Answer 2

[Y]

a) Dễ thấy x = 0 không là nghỉ=ệm của pt đã cho

Chia cả 2 vế của pt cho \(x^2\ne0\) ta đc :

\(2x^2-21x+74-\frac{105}{x}+\frac{50}{x^2}=0\)

\(\Leftrightarrow2\left(x^2+\frac{25}{x^2}+10\right)-21\left(x+\frac{5}{x}\right)+54=0\)

\(\Leftrightarrow2\left(x+\frac{5}{x}\right)^2-21\left(x+\frac{5}{x}\right)+54=0\)

\(\Leftrightarrow2t^2-21t+54=0\) ( với \(t=x+\frac{5}{x}\) )

\(\Leftrightarrow\left(2t-9\right)\left(t-6\right)=0\)

\(\Leftrightarrow\left[{}\begin{matrix}t=\frac{9}{2}\\t=6\end{matrix}\right.\Leftrightarrow\left[{}\begin{matrix}x+\frac{5}{x}=\frac{9}{2}\\x+\frac{5}{x}=6\end{matrix}\right.\)

\(\Leftrightarrow\left[{}\begin{matrix}x^2-\frac{9}{2}x+5=0\\x^2-6x+5=0\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}\left(x-\frac{9}{4}\right)^2=\frac{1}{16}\\\left(x-1\right)\left(x-5\right)=0\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}x-\frac{9}{4}=\frac{1}{4}\\x-\frac{9}{4}=-\frac{1}{4}\\x-1=0\\x-5=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}x=\frac{5}{2}\\x=2\\x=1\\x=5\end{matrix}\right.\) ( TM )

Vậy tập nghiệm của pt đã cho là \(S=\left\{\frac{5}{2};2;1;5\right\}\)