His Guide to Everything Quadratic has finally answered my burning question about the term "quadratic":
The prefix quad- means “four” and quadratic expressions are ones that involve
powers of x up to the second power (not the fourth power). So why are quadratic
equations associated with the number four?
Answer: These equations are intimately connected with problems about squares and quadrangles. (In fact, the word quadratic is derived from the Latin word quadratus for square.) Questions about quadrangles often lead to quadratic equations. For example, consider the problem:
A quadrangle has one side four units longer than the other. Its area is 60 square units. What are the dimensions of the quadrangle?
If we denote the length of one side of the quadrangle as x units, then the other must be x+4 units in length. We must solve the equation: x(x+4) = 6, which is equivalent to solving the quadratic equation x^2 + 4x - 60 = 0.
Solving quadratic equations, even if not derived from a quadrangle problem, still
involves the geometry of four-sided shapes. As we shall see, all such equations can
be solved by a process of “completing the square.”