Quicksort to find median?
Quicksort is for sorting, the algorithm you refer to is a selection algorithm known as Quick Select.


Since you can only pick as pivot a number that is in the list case #1 never happens.
Worst case is that you continually pick a number that partitions the list into 2 lists: A list with just one element and a list with $n-1$ elements, if this is the case in each iteration you only rule out a single element.


So first iteration you do $n$ operations, second iteration $n-1$ operations, third one
 $n-2$ operations, ... last iteration 1 operation.

Which its the sum of the first $n$ natural numbers:

$1+2+3+... +(n-2)+(n-1)+ n = n(n-1)/2$ operations = $O(n^2)$