Can the quadratic formula be explained intuitively? Most people know what the quadratic formula is so I won’t post it here (Besides, I don’t know how to properly post formulas in general).
I was wondering if there is an intuitive explanation as to why the quadratic formula is structured the way it is. 

The development of the quadratic formula is based on solving the quadratic equation in the form $$ax^2+bx+c=0.$$
By completing the square, we have
\begin{align}
ax^2+bx+c&=0\\
ax^2+bx&=-c\\
x^2+\frac{b}{a}x&=-\frac{c}{a}\\
x^2+\frac{b}{a}x\color{blue}{+\left(\frac{b}{2a}\right)^2}&=\color{blue}{\left(\frac{b}{2a}\right)^2}-\frac{c}{a}\\
\left(x+\frac{b}{2a}\right)^2&=\frac{b^2-4ac}{4a^2}\\
x+\frac{b}{2a}&=\pm\sqrt{\frac{b^2-4ac}{4a^2}}\\
x&=-\frac{b}{2a}\pm\frac{\sqrt{b^2-4ac}}{2a}\\
x&=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
\end{align}
