system
Given the following post, respond as if you're commenting on Reddit:
human
Completing the Square: Why does it give the turning point?

Why is it when you turn a quadratic into the form of completing the square, why does this give the turning point?
gpt
It's all to do with translations.

If you've got some function, f(x), you can translate it (i.e. move it) a units to the right by *subtracting* a from every x value, f(x - a).

Likewise, you can translate it b units up by *adding* b to f(x), f(x) + b.

Both translations can be compounded, giving us a translation of (a, b) and a function of f(x - a) + b.

So, to answer your question, by completing the square we write a quadratic into a form that we can use to determine the minimum point based on the translation that has taken place on it:

If f(x) = x², we can translate it with the same function above, so, f(x - a) + b = (x - a)² + b, which you might recognise as the completed square form.

Now, how does this tell us where the minimum is? We know that the minimum of f(x) = x² is (0, 0), so by translating that by (a, b), all coordinates get shifted by the vector quantity (a, b), so we end up with a minimum of (0 + a, 0 + b), or simply (a, b).

Hope that all makes sense!