I've just learned that it is posible to rotate the axes $x$ and $y$ to obtain the axes $x'$ and $y'$ such that the quadratic form $$ax^2+bxy+cy^2$$ converts to $$\lambda _1x'^2+\lambda _2y'^2$$ So, is it possible to do the same to the form $$ax^3+bx^2y+cxy^2+y^3$$?
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$\begingroup$ This is essentially solving the cubic equation. You will have to depress the cubic and make Vieta's substitution. $\endgroup$– user7777777Commented Jun 23 at 17:07
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