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 equation $$ax^2+bx+c=0$$ can be normalized to a simpler form by using a linear change of variable such as
$$x=pt+q.$$
Plugging in the equation, we get
$$ap^2t^2+(2apq+bp)t+aq^2+bq+c=0.$$
Now (WLOG $a>0$) we are free to set
$$\begin{cases}ap^2=1,\\2apq+bp=0\end{cases}$$
and the equation simplifies to
$$\color{green}{t^2-d=0}$$ for some constant $d$.
Obviously the solutions are
$$t=\pm\sqrt{d}$$ and are real for $d>0$.
Hence the solutions in $x$ are a linear function of $\pm\sqrt d$. Solving for the parameters $p, q, d$, you obtain the classical formulas.
