Solve $\begin{cases}u+(u+1)^2=20\\u^2+(u+1)=20\end{cases}$ This question is related to the previous question I posted today. So I am trying to solve $$\begin{cases}u+(u+1)^2=20\\u^2+(u+1)=20\end{cases}.$$
The system is equivalent to $$\begin{cases}u^2+3u-19=0\\u^2+u-19=0\end{cases}.$$ We can actually solve the two equations for $u$ and see if they have common solution(s). But I tried to substract the the two equations to get $$2u=0\\ \Rightarrow u=0.$$ Actually $u=0$ isn't root to any of the equations. Is this a contradiction? What do we actually get when we substract the equations? (or when we substract 2 equations in general) What does this mean?

You've showed that the only possible solution to your system of equations is $u=0$. But $u=0$ is not in fact a solution. So there are no solutions.
Since this is a system of two equations in only one variable, you should not find this fact surprising! You'd have to get very lucky in order for the system to have any solutions.
