What is completing the square? Why is it called, completing the square? Is square metaphorical in this sense? How do you complete the square and what is it used for?
Thank you, regards.

Consider the quadratic term $x^2+2x-11$. The goal of completing the square is to write this term in a form where $x$ appears only once and the idea behind this is to use the binomial formula $(a+b)^2=a^2+2ab+b^2$ in opposite direction. So we set $a=x$ and then it looks like $2ab$ should be $2x$ and since $a=x$ we get $2xb=2x$ which gives us $b=1$. Then
$$(x+1)^2=x^2+2x+1^2=x^2+2x+1.$$
But wait a minute, we want $x^2+2x-11$. What do we do? We add the missing $1$ to complete the square:
\begin{align}
x^2+2x-11 &= x^2+2x+1-1-11 \\
&= (x+1)^2-1-11 \\
&= (x+1)^2-12.
\end{align}
