<s>[INST] 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. 
 [/INST] 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}
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