Cardano after the depressed form

Problem

Reduce to the depressed form the equation $x^3+6x^2+9x+4=0$ and apply Cardano’s formula.

Solution

With $a=1$, $b=6$ we set $x=y-\dfrac{b}{3a}=y-2$. Substituting: $(y-2)^3+6(y-2)^2+9(y-2)+4 = y^3-3y+2 = 0,$ hence $p=-3$, $q=2$. The discriminant is $D=\Bigl(\frac{q}{2}\Bigr)^2+\Bigl(\frac{p}{3}\Bigr)^3 = 1 + (-1) = 0.$ With $D=0$ Cardano’s formula gives $y=\sqrt[3]{-q/2}+\sqrt[3]{-q/2}=2\sqrt[3]{-1}=-2$, that is $x=y-2=-4$. The other (double) roots are obtained by dividing: $y^3-3y+2=(y+2)(y-1)^2$, whence $y=1$ (double) and therefore $x=-1$ (double). In summary $\boxed{x=-4 \text{ (simple)},\ x=-1 \text{ (double)}}$.

Links

Topics: Quadratic equations
Concepts: Cubic equation · Depressed form
Methods: Cardano cubic
Skills: Solving equations
Exercise type: Solving an equation