This explanation would only be good for those that already understand the subject. One struggles to work through the hidden assumptions: the explanation is there with a few pauses and external research. It is not an introduction; thus, the title is click-bait.
Do 1 and 4 count as one (pair), or as 2 separate solutions? I.e., do n and p-n count as one solution, or as two? I.e., do you only need to find the first residue to find the other, or do you need to find the first two residues to find them all?
The question of whether C is a quadratic residue doesn't depend on the actual solution to the congruence x^2 ≡ C. If at least one solution exists, then C is a quadratic residue.
Here is a simpler proof of this fact which viewers might like. :) We know x² ≡ (p - x)² ≡ a (mod p). Let y = p - x. Then p | x² - y² = (x - y)(x + y). Now, p is a prime so p | x - y or p | x + y (Euclid's lemma) and therefore x = y or x + y = p because x, y ∈ {1, 2, ..., p - 1} which proves the claim. Glad to help!
