For which values of the parameter $p$ does the following equation have exactly one solution?

$$
x+1=\sqrt{p x}
$$
Considering the domain of interpretation and the range of values on the right side, on the one hand $p x \geq 0$, and on the other hand $x+1 \geq 0$. Based on this, if $p>0$, then $x \geq 0$, and if $p>0$, that is, $p>4$ or $p<4$; in the case of $p>4$, this means that the roots of equation (2) are either both roots of (1) as well, or neither of them is, so these values of $p$ cannot be considered.

If $p<0$, then the sum of the two roots of (2), $p-2$, is negative, so both roots are negative. Since they are different and their product is 1, one is less than -1 and thus not a root of (1), while the other falls between -1 and 0, and is therefore a solution to (1). In the case of $p<0$, therefore, one of the two different roots of (2) - the one with the smaller absolute value - is also a root of (1).

Thus, equation (1) has exactly one root when $p=4$ and when $p \leq 0$.

Remarks. 1. If we plot the graphs of the functions on both sides of (1), the appropriate values of $p$ can be easily read off.

2. In more than three-quarters of the papers, only the possibility of $D=0$ was clarified. It is telling that so many are satisfied with a template recognition instead of trying to understand what the problem is actually about.