\(\left\{{}\begin{matrix}x^2+2y^2=3\\x+y=m+1\end{matrix}\right.\) <=> \(\left\{{}\begin{matrix}\left[\left(m+1\right)-y\right]^2+2y^2=3\\x=\left(m+1\right)-y\end{matrix}\right.\) <=>\(\left\{{}\begin{matrix}\left(m+1\right)^2-2\left(m+1\right)y+y^2+2y^2=3\left(1\right)\\x=\left(m+1\right)-y\end{matrix}\right.\)
Hệ PT có nghiệm duy nhất <=> (1) có nghiệm duy nhất <=>\(\Delta'=0\)
<=> \(\left(m+1\right)^2-3\left[\left(m+1\right)^2-3\right]=0\)
<=> \(9-2\left(m+1\right)^2=0\)
<=> \(\left(m+1\right)^2=\dfrac{9}{2}\)
<=> \(\left[{}\begin{matrix}m+1=\dfrac{3\sqrt{2}}{2}\\m+1=-\dfrac{3\sqrt{2}}{2}\end{matrix}\right.\) <=> \(\left[{}\begin{matrix}m=\dfrac{3\sqrt{2}-2}{2}\\m=\dfrac{-3\sqrt{2}-2}{2}\end{matrix}\right.\)