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. 

The rate at which the area is changing $dA/dx$ is $p/2 - 2x$ and has a critical value at either $0$ or when $x$ is $1/4p$ so if $2 x$ values are used to preserve the parallels of the rectangle the remaining $p/2$ is the $y$ values which are just $x$ again.
