Why is it called a quadratic if it only contains $x^2$, not $x^4$? Should it not be bidratic or didratic etc?
The Latin prefix quadri- is used to indicate the number $4$, for example, quadrilateral, quadrant, etc. However, it also very commonly used to denote objects involving the number $2$. This is the case because quadratum is the Latin word for square, and since the area of a square of side length $x$ is given by $x^2$, a polynomial equation having exponent two is known as a quadratic ("square-like") equation. By extension, a quadratic surface is a second-order algebraic surface.
By analogy, since the volume of a cube of side length $x$ is $x^3$, a polynomial equation having exponent three is called a cubic equation. An equation of degree four is then unimaginatively called a quartic equation, or sometimes (more commonly in older sources) a biquadratic equation.