The depressed form

Cardano’s formula for third-degree equations is the natural generalisation of the $\Delta$ formula for the second degree, and it is the real reason why the mathematicians of the Renaissance had to surrender to the existence of imaginary numbers (which we shall meet formally in the chapter on complex numbers). The first step consists in simplifying the equation by removing the second-degree term.

Step 1: reduce to the depressed form. Every third-degree equation $ax^3+bx^2+cx+d=0$ ($a\neq 0$) can be reduced, with the substitution $x=y-\dfrac{b}{3a}$, to the depressed form $y^3 + py + q = 0,$ in which the second-degree term disappears. It is the analogue of the completing of the square done for the second degree.

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Topics: Second-degree equations
Concepts: Third-degree equation · Depressed form
Skills: Solving equations
People: Gerolamo Cardano