Say you have a function 0 = ax^2 + bx + c, where a = 1, just like the binomial you suggested.
To complete the square, you must add 25. Why? But also add it to the other side!
If you take half of the second term (bx, where b = 10), then you've got 5x. To complete the square you square half of b, or (.5b)^2, so you know what you need to add to complete the square. Once you've added 25, you can then factor the trinomial 0 + 25 = x^2 + 10x + 25 into 25 = (x+5)^2, or 0 = (x+5)^2 - 25. -> y = (x+5)^2 - 25.
This can be completed where b is not an even number or an integer (b can also be a fraction!)
Let's try a case where a doesn't equal one, say, where a = 2, b = 10, and c = 9 -> 0 = 2x^2 + 10x + 9
Factor out the 2 from only the first two terms (important): 0 = 2(x^2 + 5x) + 9
Now, we take 1/2 of b, and then square it, so we know what to add or subtract to complete the square. But, we must add the same thing to both sides, so don't neglect the coefficient (the 2 we pulled out earlier). You can also subtract the same value on the same side of the equation so you don't have to do it later.
0 + 2(25/4) = 2(x^2 + 5x + 25/4) + 9 -> 25/2 = 2(x + 5/2)^2 + 9 -> 0 = 2(x + 5/2)^2 - 7/2 ->
y = 2(x+5/2)^2 - 7/2,
This is very simple and methodical, and I hope I helped you!
I am a high-school Calculus student!