Tìm $n$ để hệ sau đây có nghiệm duy nhất:
         $\begin{cases}x^2+(2n+1)x+n^2+n-2=0 \\ x^4-5x^2+4<0 \end{cases}$
Giải
   Xét phương trình: $x^2+(2n+1)x+n^2+n-2=0   (1)$
Ta có: $\Delta=(2n+1)^2-4(n^2+n-2)=9$
Vậy $(1) \Leftrightarrow x=\frac{-(2n+1)\pm 3}{2} \Leftrightarrow \left[ \begin{array}{l}x = -n+1\\x = -n-2\end{array} \right.$
   Xét bất phương trình: $x^4-5x^2+4<0 \Leftrightarrow 1<x^2<4$
                $\Leftrightarrow |x|>1$ và $ |x|<2 \Leftrightarrow -2<x<-1 \vee 1<x<2$
Hệ có nghiệm duy nhất $\Leftrightarrow \left[ \begin{array}{l}-n+1\in (-2;-1)  và -n-2 \notin (1;2)\\-n+1\in (1;2)  và -n-2 \notin (-2;-1) \\  -n-2\in (-2;-1)  và -n+1 \notin (1;2) \\  -n-2\in (1;2)  và -n+1 \notin (-2;-1)\end{array} \right.$
$\Leftrightarrow \left[ \begin{array}{l}(-2<-n+1<-1)\wedge (-n-2 \leq 1 \vee -n-2 \geq 2)\\(1<-n+1<2)\wedge (-n-2 \leq -2 \vee -n-2 \geq-1) \\  (-2<-n-2<-1) \wedge (-n+1 \leq 1 \vee -n+1 \geq 2) \\  1<-n-2<2 \wedge (-n+1 \leq -2 \vee -n+1 \geq -1)\end{array} \right.$
$\Leftrightarrow \left[ \begin{array}{l}(2<n<3)\wedge (n \geq -3 \vee n \leq -4)\\(-1<n<0)\wedge ( n \geq 0 \vee n \leq -1 (VN)\\ (-1<n<0) \wedge (n \geq0 \vee n \leq -1)   (VN) \\  (-4<n<-3)\wedge (n \geq 3 \vee n \leq 2  \end{array} \right.\Leftrightarrow \left[ \begin{array}{l}2<n<3\\-4<n<-3\end{array} \right.$

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