Quadratic is defined to be, "involving the second and no higher power of an unknown quantity or variable."
The word Quadratic always throws me off, since the number 4 immediately comes to mind, because of the Quad. Why was the name Quadratic chosen? Why couldn't something like 'Bi-atic equation be chosen'
Supposedly, an equation with the highest degree of 4 is called, quartic, which is again not very intuitive.
When you compute the area of a square, you raise the edge at the power $2$. This is where the term "squaring a number" came from. Since "quad", as you correctly say, is the Latin root for "four", and a square has four edges, a squared algebraic expression became a synonym of for a quadratic algebraic expression.