The van der Waals (from his thesis of 1873) equation is a cubic in the molar volume. Since this is the first equation of state studied in chemistry more complicated than the ideal gas equation, it is noteworthy that the solution of this cubic is rarely if ever addressed. When (lowering the temperature) the two imaginary roots coalesce with the real root of the van der Waals equation we have the critical point. At Kelvin temperatures below Tc. the equation has three real roots; we explore obtaining those roots to aid in the Maxwell construction required to obtain the vapor pressure. These notes are intended to aid the reader whose long-unused calculus training has atrophied through professional neglect or for students whose comfort level with calculus is minimal, hence the long tedious algebraic manipulations
David, Carl W., "The van der Waals Equation as a cubic" (2015). Chemistry Education Materials. Paper 88.