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Using clever substitutions to minimize the number of symbols in the calculation of the roots of the general cubic equation

We should bring the equation into the depressed cubic form $$y^3+py+q=0$$ by $y=x-\frac{b}{3a}$ substitution as Alex did. Then, an interesting idea is the substitution $y=\frac{2i\sqrt p}{\sqrt3}\sin\...
Bob Dobbs's user avatar
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Necessary and sufficient conditions that a cubic equation has three positive real roots

We will prove that if (2),(3) and (4) hold, the three roots must be all positive. For $x⩽0$, (2),(3),(4) imply that $x^3⩽0,px^2⩽0,qx⩽0,r<0$, so $x^3+px^2+qx+r<0$, so $x^3+px^2+qx+r$ have no ...
hbghlyj's user avatar
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Using clever substitutions to minimize the number of symbols in the calculation of the roots of the general cubic equation

One can use the following : The solutions of $ax^3+bx^2+cx+d=0\ (a\not=0)$ are $$\begin{align}x_1&=-2A-\frac ba \\\\x_2&=A+Bi \\\\x_3&=A-Bi\end{align}$$ where $$\begin{align}A&=\frac{B}...
mathlove's user avatar
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Using clever substitutions to minimize the number of symbols in the calculation of the roots of the general cubic equation

I studied the following general approach to solve a cubic equation $ax^3+bx^2+cx+d=0$ with $a\ne 0$. Put $x=y-\frac b{3a}$. Then $y^3+py+q=0$, where $p=\frac{3c-b^2}{3a^2}$ and $q=\frac{2b^3-9abc+27a^...
Alex Ravsky's user avatar
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Prove that $a=0$ if and only if $b=0$ for the cubic $x^3 + ax^2 + bx + c=0$ whose roots all have the same absolute value.

I'm proving what @Joshua Woo had written. I think you'll be able to continue from there. $$pq+qr+rp=0$$ then $$2pq+2qr+2rp=0$$ This means that $$(p+q+r)^2=p^2+q^2+r^2+2pq+2qr+2rp=p^2+q^2+r^2$$ Hence $(...
Gwen's user avatar
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Prove that $a=0$ if and only if $b=0$ for the cubic $x^3 + ax^2 + bx + c=0$ whose roots all have the same absolute value.

The problem given is equivalent to the following situation: Given $|p|=|q|=|r|$, prove that $p+q+r=0$ if and only if $pq+qr+rp=0$. ($\rightarrow$) As there are only three variables, the number of ...
Joshua Woo's user avatar
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Does an invertible real matrix have a real cubic root

For square roots of real $A$ to exist, it is enough that all of the real eingenvalues of $A$ are $>0$. Cubic roots, and more generally roots of odd order, exist for every real invertible $A$. The ...
orangeskid's user avatar

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