Question

I was wondering if anyone could give me a link to a video or a website that gives proof and great intuition on why completing the square works?

I can complete the square but I do not understand why it works and it feels empty not understanding why it works.

Answer 1

Well I can't really send you a video or link of WHY it works, but if you want to know why it works, let's see if I can explain it. The reason it works is because it involves the factoring of perfect square trinomials. For example, if you have x^2 + 8x + 16 = 0, you would got through the steps.
x^2 + 8x + 12 = 0
x^2 + 8x = -12
At this point, I hope you recognize that the left side is a part of a perfect square. It's just missing the third terms or constant. This is something that you learn to do in higher levels of math. It's called manipulating the problem. You have to move things around in order to simplify further. This is kind of like rationalizing the denominator. Next, you would add a 16 in order to "complete the square" or perfect square trinomial.
x^2 + 8x + 16 = -12
Now, this wouldn't work if you didn't also add a 16 on the other side because then the value of "x" would change. However, if you do things on both sides of the equation, then the value stays the same. This is called balancing the equation.
x^2 + 8x + 16 = -12 + 16
x^2 + 8x + 16 = 4
Now, it's looking like a perfect square trinomial on the left side. You know how to factor one of these, so the result is (x + 4)^2 = 4. Then, you also have to bring 4 to the other side since it is a quadratic. (x + 4)^2 - 4 = 0.
Hopefully this was helpful.