Intuitive understanding of maximum value of quadratic function In trying to understand why the maximum area of a rectangle with a fixed perimeter occurs when the base is equal to the height, I got as far as this expression:
$A = (p/2)x - x^2$
from
$p = 2x + 2y,x + y = p/2, y = p/2 -x, A = x(p/2 - x)$ 
I know that I need to find the maximum value of the top expression, which can be generalised to $bx - x^2$
My question is, is there an intuitive (possibly visual) way to understand how to find the greatest possible value of this expression? Ideally I'd like to avoid anything but the most basic algebraic steps, and not have to refer to graphs or the quadratic equation. 

A quadractic equation 
$$
q(x) = a x^2 + bx + c
$$ 
with non-zero coefficient $a$ has either a minimum or a maximum, depending on the sign of $a$. 
If you plot the graph you will see the typical parabola shape.
One way to determine the extremum is to bring $q$ into the form
$$
q(x) = a (x - S)^2 + T
$$
where $(S, T)$ are the coordinates of the extremal point (vertex).
In your case we have
\begin{align}
A(x) &= -x^2 + (p/2) x \\
&= -(x^2 - (p/2) x) \\
&= -((x - p/4)^2 - (p/4)^2) \\
&= -(x- p/4)^2 + (p/4)^2
\end{align}
from which you can read the coordinates of the extremum.
