I’m confused about the third step In the proof of the quadratic equation with the form $$ ax^2 + bx + c = 0. $$ In the third step it is added $\bigl(\frac{b}{2a}\bigr)^2$ to both sides to complete the square, and it would make sense if it was geometrically possible.
Screenshot of derivation of quadratic formula.
I mean, if it was $ax^2 + bx - c = 0$, that is, $ax^2 + bx = c$, we can think of it as a square with both sides $x$, and a (maybe) rectangle with side $\frac{b}{a}$ and the other $x$, and they have an area of $\frac{c}{a}$.
so we can complete that square, and if we continue the rest of following algebra steps, we get a quite similar result that is, $$ x = \frac{-b±\sqrt{b^2+4ac}}{2a} $$ So... Did they just apply the concept of completing the square to find a square with a negative area? is a negative area possible?