# Questions tagged [cubics]

This tag is for questions relating to cubic equations, these are polynomials with $~3^{rd}~$ power terms as the highest order terms.

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### Are there any ways to convert inverse trigonometric values to radicals?

When we solve a cubic equation $ax^3+bx^2+cx+d=0$, the roots are supposed to be in the form of radicals in real numbers or complex realm. However, if the discriminant is less than 0, the solution is ...
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### Solving Vieta's jumping problem (1988 IMO problem 6) with a cubic polynomial

I have recently watched this Numberphile video explaining how Zvezdelina Stankova solved the notorious 1988 IMO problem 6 as a student. (Most of you will know this problem) Let $x$ and $y$ be ...
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### Regular heptagon coordinates from a cubic field

Find coordinates for a regular heptagon in 3D Euclidean space where all $3$ components $(x,y,z)$ of all $7$ coordinates are elements of the same cubic field, or prove that it can't be done. Background:...
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### On the solvable octic $x^8-44x-33 = 0$ and the tribonacci constant

I had discussed the solvable octic trinomial, $$x^8-44x-33=0\tag1$$ way back in this old MSE post, but I revisited this inspired by another solvable octic, $$y^8-y^7+29y^2+29=0\tag2$$ which I also ...
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### How is $\xi+2a\eta<0$ an "obvious necessary condition" for $y^3+2y^2(1-2a-\xi)+y(1-4\xi+8a\xi)-2\xi-4a\eta >0$ to be satisfied for positive $y$?

How is $$\xi+2\alpha\eta<0$$ an 'obvious necessary condition' for the inequality $$y^3+2y^2(1-2\alpha-\xi)+y(1-4\xi+8\alpha\xi)-2\xi-4\alpha\eta >0$$ to be satisfied for positive $y$ (as claimed ...
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### Solving Cubic Systems of Diophantine Equations

What techniques are there for solving systems of Cubic Diophantine equations? I know there is no general purpose technique and looking at some papers it can quickly go over my head even for just a ...
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### Will the perpendicular bisector between the line connecting two cubic roots of the same arc never intersect its turning point?

The quadratic graph: $$f(x) = (x+2)(x+1)$$ would have a midpoint between its roots at $x = -1.5$. This line would intersect its turning point. However the cubic graph: $$f(x) = (x+1)(x-2)(x+3)$$ ...
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### Cubic Polynomial with leading coefficient 1 satisfying certain condition

I did translation which may differ from original question The cubic function $f(x)$ with the coefficient of the leading term equal to 1 and when a function $g(x)$ that is continuous on a set ...
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### Further information on the reduction of cubic equations to a system of two conic sections

This question follows on from one I have previously asked, How to separate cubic equations into two conic sections: Deep dive into Omar Khayyam and I now would like some further advice on some ...
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### Estimate root scale of a cubic equation

Consider the following cubic dispersion equation of $\omega(k)$ $$\omega^2-\omega_a^2(k)-\frac{\alpha^2k^4}{\omega-\omega_b(k)}=0.$$ $\omega_a=vk,\omega_b=b-v'k$ are two unhybridized dispersions that ...
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### Is the theory of $\Delta\le0$, true in cubic functions$?$

If $ax^2+\frac{b}{x}\ge c$ $\forall x>0$ where $a>0 \:\: , b>0$ Show that $27ab^2\ge4c^3$ My work: Let a function $f(x)$ be $$ax^2+\frac{b}{x}\ge c$$ or we can rewrite $f(x)$ as ax^3-cx+b\... ### About extreme values of $\{f(x)-x\}^2$ when $f(x)$ is a cubic function.
$t \ge 6$, $t \in \mathbb{R}$ $f(x) = \frac{1}{t}\left( \frac{1}{8}x^3 + \frac{t^2}{8}x+2\right)$ $\{f(x)-x\}^2$ has an extreme value on $x = k$ Sum of such $k = g(t)$ $g(p) = -1$ for some \$p \in \...