Suppose we wish to make a rectangle with a given area and perimeter. This is an interesting problem! Does the number of possible answers depend on the specific area and perimeter? Certainly! It all comes down to thinking about squares, since squares maximize the area given a fixed perimeter. If the area of a square with the given perimeter is LESS than the desired area, then there's no way we can make such a rectangle. If the area of the square is equal to the desired area, then that's our only answer! Now, how about if our square's area is larger than the desired area? We'll get two possible different lengths of a given side of the rectangle - one representing the rectangle's width, and the other representing it's height. Or we could also think of them as two different rectangles - a tall one, and its rotation by a quarter turn (which makes it wide). By symmetry, we know that the difference between the square's side length and the shorter side will be the same as the difference between the square's side length and the longer side. How large is that difference? Exactly enough to diminish our shape's area from the square's area to the desired area. And that difference in length is simply the square root of the difference between the square's area and the desired area!
It would have been better with pictures :). Anyway, the quadratic formula is probably the greatest mistake in all of mathematics education. Somehow we use the word "quadratic" and even the phrase "complete the square," but never have I ever seen someone draw the said square!
While I think notation is often great for expressing ideas concisely and precisely, I think an excess of notation not a good way to communicate concepts. Nobody should memorize the quadratic formula! We should understand instead how to think about areas and lengths, and then we solve the problems in quadrature that we want.