Derive quadratic formula

Question

I cannot understand how quadratic formula to solve for $x$ was derived.

On this website, it explains the steps

Following I understand

but I cannot understand how they got

$b/2a$

and why they are squaring it

$b^2/4a^2$

Really, I am baffled!

Answer 1

$x^2+\frac{b}{a}x+K^2=(x+K)^2$

$x^2+\frac{b}{a}x+K^2=x^2+2xK+K^2$

$\frac{b}{a}x=2xK$

$k=\frac{b}{2a}$

Then we know that

$x^2+\frac{b}{a}x+(\frac{b}{2a})^2=(x+\frac{b}{2a})^2$

$x^2+\frac{b}{a}x+\frac{b^2}{4a^2}=(x+\frac{b}{2a})^2$

This process is called completing the square. Now we can factor a perfect square. Is this answer sufficient?

Answer 2

Hint:

This is the completing the square trick. Usefull in many other cases and simple to visualize.

The black square is $x^2$, the red rectangle is $bx$, so $x^2+bx$ can be transformed to a square adding the little blue square, that is $(b/2)^2$.