<s>[INST] Why is the sum of the roots of $x^3+9=12x$ zero? I found the roots of the equation to be 
$$x=3,\frac{\sqrt{21}}{2}-\frac{3}{2},-\frac{\sqrt{21}}{2}-\frac{3}{2}$$
Algebraically, why is the sum of these roots equal to zero?
 [/INST] Consider the following cubic form:
$$ax^2+bx^2+cx+d=0$$
For your equation:
$$1\cdot x^3+0\cdot x^2-12\cdot x+9=0$$
Using Vieta's formulas, the sum of roots is equal to $-\frac{b}{a}$.
$$-\frac{0}{1}=0$$
Thus the sum of roots is $0$.
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