Can you find the maximum or minimum of an equation without calculus? Without using calculus is it possible to find provably and exactly the maximum value
or the minimum value of a quadratic equation 
$$ y:=ax^2+bx+c $$
(and also without completing the square)?
I'd love to know the answer.

One approach for finding the maximum value of $y$ for $y=ax^2+bx+c$ would be to see how large $y$ can be before the equation has no solution for $x$.
First rearrange the equation into a standard form:
$ax^2+bx+c-y=0$
Now solving for $x$ in terms of $y$ using the quadratic formula gives:
$x= \frac{-b\pm \sqrt{b^2-4a(c-y)}}{2a}$
This will have a solution as long as $b^2-4a(c-y) \geq 0$
You can rearrange this inequality to get the maximum value of $y$ in terms of $a,b,c$. See if you get the same answer as the calculus approach gives. Remember that $a$ must be negative in order for there to be a maximum.
