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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Add constraints to cubic or quintic polynomial [closed]
How can I add constraints to cubic or quintic polynomial such that the generated line is within a region. For example in the blue colored region below:
Image_Graph
For example if I generate a line ...
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Why do cubic equations always have at least one real root, and why was it needed to introduce complex numbers?
I am studying the history of complex numbers, and I don't understand the part on the screenshots. In particular, I don't understand why a cubic always has at least one real root.
I don't see why the ...
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Are the 28 bitangents on quartic curves bounded distance away from each other?
We know that the 28 bitangents on a smooth plane quartic curve over $\mathbb{C}$ are all distinct. Are the bitangents bounded distance away from each other? More precisely, is there a constant $d>0$...
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Question about real inflection points of cubic curves in P^2(C)
An elementary question about real inflection points of cubics:
Textbooks mention that non-singular cubics in $P^2(C)$ have 3 real and 6 complex inflections and show the Hesse normal form $ x^3 + y^3 +...
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Quicker and non-trivial methods for solving Cubic Equation
Motivation : There have been many elementary ways like Hit-and-trial method, Polynomial division and others used in teaching how to solve cubic equation. I wanted to find a method that is faster to ...
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What is a curve at $y=\infty$ mean?
On the wikipedia for the trident curve, $xy+ax^3+bx^2+cx=d$, two graphs are shown:
Both are for the case where $a=b=c=d=1$, with the first matching what I find in desmos, but the latter being the '...
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Why is the locus of points given two segments AB and CD such that APB=CPD give a degree-3 curve (in complex proj plane)?
The isoptic cubic is defined as the locus of points given two segments AB and CD (similarily oriented) such that APB=CPD (directed angles). By elementary geometry this would go through AB $\cap$ CD, ...
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Proof of Thomson cubic pivotal property without coordinates
The Thomson cubic is defined as the cubic going through A,B,C, the three side midpoints, the three excenters. Is there a way to prove its pivotal property (any two isogonal conjugates on it have a ...
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Can axes $x$ and $y$ be rotated to eliminate the crossed product terms in a cubic form?
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 ...
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Solving a mixed system of 2 cubic and 2 quadratic equations with 4 unknowns
I tried plugging these cubic and quadratic equations into Wolfram Alpha and Symbolab but both said the same thing, too much computing time required. Now I am struggling to solve these equations and I ...
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For cubic surface, if $\dim(\operatorname{Sing}(X))\geq 1$ then a line is contained a in $\operatorname{Sing}(X)$
Let $X$ to be an irreducible cubic surface in $\mathbb{P}^3_{\mathbb{C}}$. Is it true that if $\dim(\operatorname{Sing}(X))\geq 1$, then a line is contained in $\operatorname{Sing}(X)$? I.e, does a ...
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Is there an isotomic analogous of circular points of infinity?
In isogonal pivotal (with pivot at the line of infinity) cubics with respect to a triangle $\triangle ABC$. By a suitable projective transformation, fixing $A$,$B$, and $C$, sending the incenter to ...
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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.
Take three real numbers $a, b$ and $c$ such that the roots of equation $x^3+ax^2+bx+c=0$ have the same absolute value. We need to show that $a=0$ if and only if $b=0$.
I tried taking the roots as $p, ...
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Show that if $x=-1$ is a solution of $x^{3}-2bx^{2}-a^{2}x+b^{2}=0$, then $1-\sqrt{2}\le b\le1+\sqrt{2}$
$$x^{3}-2bx^{2}-a^{2}x+b^{2}=0$$
Show that if $x=-1$ is a solution, then $1-\sqrt{2}\le b\le1+\sqrt{2}$
I subbed in the solution $x=-1$, completed the square, and now I'm left with the equation $\...
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Calculate on which side of a cuboid is a given point located?
Correct me if I'm using incorrect terms, I'm not well-versed with geometrical terminology
I'm writing a code in which
I have a point &
I want to identify if the points lies on front, back, top, ...
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