# Questions tagged [roots-of-cubics]

For questions related to roots of a cubic equation. All of the roots of the cubic equation can be found by the following means: algebraically, trigonometrically or numerical approximations of the roots.

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### Cosine of a multiple of arctangent

I would like to find an expression for the following: $$\cos\left(\tfrac{1}{3} \arctan(B/A)\right)\\ \sin\left(\tfrac{1}{3} \arctan(B/A)\right)$$ This is a problem that comes up when trying to ...
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### How to solve $x^3-3x+3=0$? [duplicate]

I have the following cubic equation. Is it possible to solve it analytically in very "easy" way to a student only has a pre-calculus level: The equation is: $$x^3-3x+3=0$$ The real root I ...
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### Three real roots of a cubic

Question: If the equation $z^3-mz^2+lz-k=0$ has three real roots, then necessary condition must be _______ $l=1$ $l \neq 1$ $m = 1$ $m \neq 1$ I know there is a question here on stack about ...
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### inverting the first three elementary symmetric polynomials

Given the first three elementary symmetric polynomials $e_1 = x_1 + x_2 + x_3$, $e_2 = x_1 x_2 + x_1 x_3 + x_2 x_3$, $e_3 = x_1 x_2 x_3$ (where $x_1, x_2, x_3 \ge 0$), I wish to solve these equations ...
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### How can we use Cardano's method to solve a real life problem?

I am making a math project for my school. We can make it on any topic, but should involve some college level math. I have chosen 'Cardano's method' as my topic. I will be showing the method to solve a ...
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### How to determine the order of the real roots of a cubic equation?

This is a self-answered question (I didn't find a reference, and thought of documenting this). Consider the equation $$t^3+pt+q=0.$$ Its discriminant is $$\Delta=-(4p^3+27q^2).$$ Suppose that it ...
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### How many roots of $x(1-x)^{2}=s$ are there in $(0,1)$?

This is a self-answered question, which is part of answering this related question. Alternative solutions are welcomed. Let $0<s < \frac{4}{27}$. Prove that the cubic equation $x(1-x)^{2}=s$ has ...
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### Need help understanding cubic formula derivation by Daniel Rui

I am reading a cubic formula derivation here: http://danielrui.com/papers/cubicPolynomial.pdf It looks fairly straight forward. The author defined: $y = \sqrt[3]{u} − \sqrt[3]v$ so far so good, I ...
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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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### How to separate cubic equations into two conic sections: Deep dive into Omar Khayyam

Lots of people have asked how to use Khayyam's method but I am studying for my dissertation so really need to understand the why. What I really don't understand/ can't find useful proofs for is how he ...
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### Is there the continuous real root of the cubic equation and is there a closed formula to present it?

For any cubic equation, $ax^{3}+bx^{2}+cx+d=0$, we know there is always a real root if $a,b,c,d$ are all real. Suppose that $a,b,c,d$ are continuous and real function with respect of $i\in \mathbb{R}$,...
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### Range of an equation $y=\frac{(x-\alpha)(x^3-3x+1)}{x-\alpha}$

If the range of $y=\frac{(x-\alpha)(x^3-3x+1)}{x-\alpha}$ is all real numbers, then number of integers in the range of $\alpha$ is (1) 2 (2) 3 (3) 5 (4)Infinite I have no idea of how to do these type ...
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### Deforming roots of a cubic polynomial

This is problem 9-1 from Milnor, dynamics of one complex variable (arxiv). Let $f_{\alpha}(z) = z + \alpha z^2 + z^3$. Show that $f_{\alpha}$ can be perturbed so that the double fixed point at the ...
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### Finding all possible roots of the equation

Find all possible solutions to the equation $$(x^3-x)+(y^3-y)=z^3-z$$ where $(x,y,z)\gt1$ and $\in\mathbb{Z}$ and not all three of them are equal. The original question didn't have the last condition ...
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$$ax^{2}+bx+c=0$$ $$x=-\frac{b}{a}+\frac{1}{\frac{b}{c}+\frac{1}{-\frac{b}{a}+\frac{1}{\frac{b}{c}+\frac{1}{...}}}}$$ $$ax^{3}+bx^{2}+cx+d=0$$ $$x=?$$ I know it can't recur like the quadratic ...
Solve the equation $64x^3-240x^2+284x-105=0$ given that the roots are in an arithmetic . I tried having the roots as $a, (a+d), (a+2d)$ Factorising out $a$, $a(1+d+2d)$