The general quartic can be brought into the reduced form
by means of the translation . If , then and .
The coordinates of the two points of inflection of are , where .
When , there are two real points of inflection and hence three real turning points. When , both points of inflection are complex and hence there is only one real turning point.
Since , , and are directly related to the geometry of the quartic, this Demonstration offers a more intuitive insight regarding how the shape of the curve is related to the coefficients of the reduced form .
The four roots of the reduced quartic equation can be expressed in terms of just three parameters, say , , , where , , are the roots of the resolvent cubic equation , known as Euler's cubic. Note that are the six roots of the resolvent sextic Thus, the key to solving the quartic is to first solve the resolvent cubic