Why does $\left(\frac b2\right)^2$ "geometrically complete the square?"

Question

I was just reading this MathisFun article on completing the square. It states that geometry can help complete the square. It starts off with a square and a rectangle (pictures come from link):

Then, it cuts $b$ in half, and moves it under the $x^2$ square:

Now, the square is "nearly completed", but it has this part that completes the square that equals $\left(\frac b2 \right)^2$ (circled in blue):

My question is, where did that part come from, and why does it equal $\left(\frac b2 \right)^2$. The article doesn't give me a reason and there are no other sources as to why.

Answer 1

$\left(x + \dfrac b2 \right)^2 = x^2 + 2\left( \dfrac b2 x \right) + \left( \dfrac b2 \right)^2 = x^2 + bx + \dfrac{b^2}{4}$