Cubic formula

The cubic formula is a very complicated formula used to solve cubics. It is not used very often, as schools don't teach it and problem writers usually hide a simpler tactic instead.

Cardano's formula

Start with the cubic $ax^3+bx^2+cs+d$ . The constant $a$ controls how steep it is. The constant $d$ shifts it up and down. The constant $c$ controls the slope of the middle. And the constant $b$ shifts it left to right. For now, let's just reduce it to $x^3+\frac{b}{a}x^2+\frac{c}{a}x+\frac{d}{a}$ , as everyone knows how to do that. Consider the cubic $(x+\frac{b}{3a})^3=x^3+\frac{b}{a}x^2+\frac{b^2}{3a^2}x+\frac{b^3}{27a^3}$ . The first two terms match, and if we substitute $x$ with $x-\frac{b}{3a}$ , we get rid of that annoying $x^2$ term. We'll just add it back at the end. Making the substitution results in: $\left(x-\frac{b}{3a}\right)^3+\frac{b}{a}\left(x-\frac{b}{3a}\right)^2+\frac{c}{a}\left(x-\frac{b}{3a}\right)+\frac{d}{a}=x^3+\frac{3ac-b^2}{3a^2}+\frac{2b^3+27a^2d-9abc}{27a^3}$ .

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Cubic formula

Cardano's formula