Questions tagged [quartics]

Equations that can be written in the form $p(x) = 0$ for a univariate polynomial $p$ of degree $4$ or $p(X_1, \ldots, X_r) = 0$ for a multivariate polynomial $p$ of total degree $4$. Questions that use this tag should usually also have the polynomial tag.

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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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Intersection of parabolas at $y = x$?

Intersection of parabolas at $y = x$? CONJECTURE: if a given point $(x,y)$ such that $x > 0$ and $y > 0$ , generates two parabolas with the $x$ axis and with $y$ axes as directrix, than I ...
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Role of normality of the Klein group in solving a quartic using Lagrange resolvents

There are several ways how a useful resolvent for solving a quartic equation can be defined, e.g. (the roots are a, b, c, d) ...
1 vote
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A new way of determining quartic for closest point on known ellipse to known point in space?

I've noticed a geometric relationship between a fixed point and the closest point on the ellipse. The equations are simplistic, I'm hoping that it is useful because of it's inherent simplicity and ...
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Star number equals product of consecutive star numbers

A star number is a number of the form $6n(n-1)+1$. They are equal to the number of dots in a star, like this: According to Wikipedia: Unique among the star numbers is 35113, since its prime factors (...
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Determine $x,y\in\mathbb{R}$ such that $t^4-2xt^2+y^2=0$ has $4$ real solutions in an arithmetic progression

Determine $x,y\in\mathbb{R}$ such that $t^4-2xt^2+y^2=0$ has $4$ real solutions in an arithmetic sequence. I'm quite stuck with this problem right here because, although I get to an answer, it isn't ...
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Convexification of $\frac{a}{12}\,(x^4-2\,x^2)+b\,x$

Let $f:\mathbb R\to\mathbb R$ $$f(x)=\frac{a}{12}\,(x^4-2\,x^2)+b\,x\,$$ with parameters $a,b>0\,$. $f$ is concave on the interval $\left[-\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}\right]$ and convex ...
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Is there an easy way to tell whether a cubic or quartic polynomial is factorable over the integers?

For a quadratic, it is easy to tell whether it is factorable. If the discriminant is a perfect square, the quadratic is factorable. Otherwise, the quadratic is not factorable. Is there anything ...
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How to find a quadratic polynomial with the information given?

Let $p(x)=x^4+ax^3+bx^2+cx+d$ (where $a,b,c,d\in\mathbb{R}$) and $p(1)=10,p(2)=20,p(3)=30$. How can I find the quartic $p(x)$ through the info given? I know how to start that is firstly considering ...
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How can you find the equation of the line that is tangent at two distinct points to the curve? [closed]

Given a curve $y = x^3-x^4$, how can I find the equation of the line in the form $y=mx+b$ that is tangent to only two distinct points on the curve? The problem given is part of the Madas Special Paper ...
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Find solutions to the polynomial $x^4+x^3-x^2-5x+4$ [closed]

This question is part of a calculus problem, involving the limit of a rational function. I modified it a little bit to make it more interesting. The question is: $x^4+x^3-x^2-5x+4=0$, find all the ...
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Solve $x\left(1+\sqrt{1-x^2}\right)=\sqrt{1-x^2}$

I would like to solve the equation $x\left(1+\sqrt{1-x^2}\right)=\sqrt{1-x^2}$ analytically, without using Wolfram Alpha. I have tried several substitutions, including $x=\cos(t)$, $\sqrt{1-x^2}=t$, ...
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Solvability criteria for a a monic almost palindromic quartic diophantine equation

My main question is: Is there a criteria for solving a quartic Diophantine equation of the form $$x^4 + ax^3 + bx^2 + ax + d = 0 \tag 1$$ We have the restriction $d \ne 1$. Here's my effort in ...
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If a hidden quadratic has no real roots, does that mean that the equation it represents also has no real roots?

Say you have the equation $y = 9x^4 + 7x^2 + 2$. There are multiple ways of finding the roots of this equation, but one of them is to let $u = x^2$, then re-write the equation as $y = 9u^2 + 7u + 2$, ...
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Show that the equation $x^4-8x^3+22x^2-24x+4=0$ has exactly two real roots, both being positive

There was a question asked on this site which was closed due to lack of showing his attempt. The question was Show that the equation $$x^4-8x^3+22x^2-24x+4=0$$ has exactly two real roots, both being ...
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Olympiad-like way to solve the equation $(x^2-4)(x^2+6x+6)=x^2-1$?

Solve the equation: $$(x^2-4)(x^2+6x+6)=x^2-1$$ I found this question from math olympiad textbook for beginners. But there is no specific hint for the solution. Is there any faster way to solve this ...
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Possible "clever" ways to solve $x^4+x^3-2x+1=0$, with methodological justification

Solve the quartic polynomial : $$x^4+x^3-2x+1=0$$ where $x\in\Bbb C$. Algebraic, trigonometric and all possible methods are allowed. I am aware that, there exist a general quartic formula. (Ferrari's ...
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Is there any transformation for the polynomial $x^4+ax^3+bx+1$ to either one of these forms $x^4+cx+d$ or $x^4+cx^3+d$ ? (like brioschi quintic form)
The polynomial I'm looking to solve is $a^4 - \alpha^4 a^2 -k^4 = 0$ The four roots of this polynomial are given as $a=\pm \lambda_1$ (pair of real roots) $a=\pm \lambda_2$ (pair of imaginary roots) ...