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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 $d>0$...
Weiyan Chen's user avatar
0 votes
1 answer
51 views

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 ...
Pedro Assumpção's user avatar
5 votes
1 answer
81 views

Is $\Bbb Q[a^2b+b^2c+c^2d+d^2a]$ the fixed field of the subgroup $\langle(1234)\rangle$ of Galois group $S_4$?

$f$ is an irreducible quartic polynomial in $\Bbb Q[x]$ with Galois group $S_4$. $a,b,c,d$ are four distinct roots of $f$. Let $K$ be the splitting field of $f$. Let $\text{Gal}(K/\Bbb Q)=S_4$. Then $...
hbghlyj's user avatar
  • 3,045
4 votes
0 answers
126 views

Galois group of a quartic, determine all intermediate subfields explicitly

Let $F$ be the splitting field of an irreducible quartic polynomial $f \in \Bbb Q[x]$. If Galois group of $F/\Bbb Q$ is $D_4$, I try to determine all intermediate subfields explicitly. $D_4=⟨σ,τ⟩$, $σ=...
hbghlyj's user avatar
  • 3,045
1 vote
1 answer
42 views

deciphering notation used by Jessop (Quartic Surfaces)

I am reading a rather famous work by Jessop "Quartic Surfaces with Singular Points". This is quite an old book published in 1916. I was wondering if someone could help me with a notation $\...
quantum's user avatar
  • 1,667
0 votes
0 answers
30 views

Can the general formula for quartic equation be simplified?

This question arises from an unusual method of solving depressed quartic $x^4 + ax^2 + bx + c = 0$ which extracts the value of each root from the product of four radical expressions. I will give an ...
George Plousos's user avatar
1 vote
1 answer
43 views

Upper bound of depressed quartic

For the depressed quartic equation $x^4-ax^2-bx-c=0$, $a,b,c>0$, is there some (relatively easy) way to find an upper bound of the positive root in terms of $a,b$ and $c$? I am aware of Ferrari's ...
Luis's user avatar
  • 81
2 votes
1 answer
57 views

A question about Dummit & Foote's explanation on the resolvent cubic and the Galois group

I am at the beginning of my study of field theory and I am reading page $\textbf{615}$ of $\textbf{Dummit & Foote}$, and the part where I have question about is shown below: Here for part $\...
ZYX's user avatar
  • 1,131
2 votes
0 answers
42 views

Does {7,4|3} have a realization on the Klein quartic?

It was pointed out to me recently that the polyhedron {7,4|3} has the same automorphism group as the Klein quartic. Specifically {7,4|3} is: $\langle \rho_0,\rho_1,\rho_2 \mid \rho_0^2, \rho_1^2, \...
Sriotchilism O'Zaic's user avatar
1 vote
3 answers
88 views

Solving The Quasi-Symmetric Quartic Equation

This is a self-answer question of the following problem: Solve the equation: $$4x^4 - 36x^3 + 61x^2 + 90x + 25 = 0.$$ See my answer.
user2661923's user avatar
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0 votes
0 answers
72 views

Solving a quartic equation without graphing [duplicate]

I'm currently working on a solution to a projectile motion problem, and I need to figure out the optimal launch angles by solving this system of equations. $15\cos(\theta)x+2\cos(\theta)=21 \\\\ (15\...
VV_721's user avatar
  • 375
1 vote
0 answers
31 views

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) ...
David Kubecka's user avatar
1 vote
0 answers
90 views

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 ...
Phedg1's user avatar
  • 147
5 votes
0 answers
201 views

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 (...
Riemann's user avatar
  • 717
4 votes
4 answers
289 views

Solve equation $(3x^2-2x-1)^2-(6x^2-4x-5)^2+x=0$

I tried to expand and I had $27x^4-36x^3-42x^2+35x+24=0$ I don't know how to solve this equation, but I sketch on graph and I have four irrational roots. Which method I need to use in this equation?? $...
Batyrbek Allamzharov's user avatar
5 votes
1 answer
176 views

Invariants of a quartic function

Some facts about cubic functions and quartic functions motivate this question: Every cubic function $f$ has exactly one inflection point $P$, and the graph $y=f(x)$ is symmetric about $P$. In ...
Joseph's user avatar
  • 580
1 vote
0 answers
91 views

How many roots in a system of quartic equations

I am considering three general tori in 3D space, each defined by a quartic equation. My primary question revolves around the number of real solutions that arise from the system of these three ...
paketecuento's user avatar
1 vote
1 answer
44 views

Erratum in Boyd and Vanderberghe p. 30?

In Boyd and Vandenberghe's "Convex optimization," page $30$, it's stated that: A related family of convex sets is the ellipsoids, which have the form $$\mathcal{E}=\{x\mid (x-x_c)^TP^{-1}(x-...
Patricio's user avatar
  • 1,604
1 vote
2 answers
50 views

Coefficient of a quartic polynomial given the conditions it's a natural number and has atleast one real root

Let $y=f(x)$ be a curve given by the solution of differential equation $dy/dx=4px^3-12x-8$ , $p ∈ N$, $f(0) = 24$ and $y=f(x)$ cuts x-axis at atleast one point. Find number of value(s) of p and sum of ...
Anon's user avatar
  • 33
0 votes
4 answers
124 views

Find $\sin{\frac{\pi}{10}}$ in form of $a+b\sqrt{k}$

Use De Moivre's Theorem to express $\cos{5\theta}, \sin{5\theta}$ in powers of $\sin{\theta}$ and $\cos{\theta}$. Show that $$\frac{\cos{5\theta}}{\cos{\theta}}=16\sin^4{\theta}-12\sin^2{\theta}+1$$ ...
J_dash's user avatar
  • 87
1 vote
2 answers
90 views

Show that the roots of the equation $b^2(x^4+1)-14b(1+b^2)x(x^2+1)+(1+196b^2+b^4)x^2=0$ are all real and all have the same sign as $b$.

If $\alpha, \beta$ are the roots of the equation $$x^2-ax+b=0\ (a,b\ real)$$ form the equation whose roots are $\frac{\alpha^3}{\beta},\frac{\beta^3}{\alpha}$, and deduce the equation whose roots are $...
J_dash's user avatar
  • 87
2 votes
2 answers
207 views

The relation between the number of solution of $f(x)=x$ and $f(f(x))=x$, when f is a quartic polynomial

I'm trying to solve the following problem : "Let $f$ be a quartic polynomial with a positive leading coefficent, and all coefficients are real numbers. Let $m$ be the number of real solution of $...
Detectives's user avatar
1 vote
1 answer
203 views

What must be true for thirteen points to impose independent conditions on quartics?

Recently I've been trying to generalize the Cayley-Bacharach theorem to the case of quartics. Here's one version of what the theorem says for cubics: Let $P_1,\dots,P_8$ be eight (distinct, closed) ...
Hank Scorpio's user avatar
  • 2,841
0 votes
1 answer
84 views

Find quartic equation combining equation of ellipse and circle

I was looking for a quatric equation that combines the equation of the circle and ellipse, which are: \begin{align*} Circle: (x - m)^2 + (y - n)^2 &= r^2 \\\ Ellipse: \frac{(x - o)^2}{a^2} + \frac{...
Erika Pan's user avatar
0 votes
0 answers
42 views

How to find smallest positive real roots of $\left(At^2+Bt+C\right)\left(Dt^2+Gt+H\right)-k^2=0$ given $A,B,C,D,G,H\in C$ and $k\in R$?

I am experimenting with quartics because of a computer program. I wanted to see if there is a simpler general way than using the quartic formula, or descartes, or ferrari methods to find the smallest ...
PlatinumFrog's user avatar
0 votes
1 answer
32 views

Efficient representations of the angular dependence of quadratic and quartic functions in 2d

Consider the set of arbitrary quadratic functions in two dimensions $F(x_1, x_2) = \sum_{ij} A_{ij} x_i x_j$, as well as the set of arbitrary quartic functions $G(x_1, x_2) = \sum_{ijkl} B_{ijkl} x_i ...
Panopticon's user avatar
1 vote
1 answer
141 views

Can the solution to $x^4+\frac{13}6Ux^3+\frac32U^2x^2+\frac13U^3x-V=0$ be written as a sum of quartic roots?

I have the following quartic polynomial equation, where $U$ and $V$ are positive real numbers: $$ x^4 + \left(\frac{13}{6} × U\right) x^3 + \left(\frac{3}{2} × U^2\right) x^2 + \left(\frac{1}{3} × U^3\...
Lawton's user avatar
  • 1,861
5 votes
8 answers
408 views

What’s the simplest way to prove that the polynomial $f(z)=z^4 + z^3 + z^2 +2z +3$ has no real zero?

I have managed to write the polynomial mentioned in the title as a sum of squares as follows: $$ z^4 + z^3 + z^2 +2z +3 = \left( z^2 + \frac12 z - \frac18 \right)^2 + \left( z + \frac{17}{16} \right)^...
A A's user avatar
  • 613
0 votes
0 answers
43 views

Can the local minima and maxima of a quartic be at any (unique) arbitrary point?

Is it possible to construct a quartic function with two local maxima and one local minima at any three arbitrary points, making some assumptions? If so, how can I do it? The x-coordinate of the local ...
Monolith's user avatar
  • 139
-2 votes
1 answer
47 views

investigated by descartes rule of sign

Use Descartes' rules of signs to discuss and determine (how many) the number of possible positive roots of each equation. $p(x)=ax^4+bx^3+cx^2+dx+e=0$
Tias F's user avatar
  • 1
3 votes
2 answers
114 views

Find the sum of the real roots of the polynomial $P(x)=(x^2+2x+4)^2+2(x^3-8)+(x-2)^2$

Find the sum of the real roots of the polynomial $$P(x) = \left(x^2+2x+4\right)^2 + 2\left(x^3-8\right)+(x-2)^2$$ If we write the polynomial as $$P(x)=(x^2+2x+4)^2+2(x-2)(x^2+2x+4)+(x-2)^2,$$ I think ...
Snowlash's user avatar
  • 115
0 votes
0 answers
74 views

How to factor quartic into quadratic factors? [duplicate]

I am having problems factoring the following quartic with quadratic roots. $$3x^4 + x^3 + 2x^2 - x + 1 = \left(x^2 + x + 1 \right) \left( 3x^2 - 2x + 1 \right)$$ All the ways I've found online till ...
VD-Flash's user avatar
5 votes
3 answers
345 views

Find the sum of the real roots of the equation $6x^4+9x^3-15x^2+9x+6=0$

Find the sum of the real roots of the equation $$6x^4+9x^3-15x^2+9x+6=0$$ We can solve the equation by dividing both sides by $x^2\ne0$. Then we'll get $$6\left(x^2+\dfrac{1}{x^2}\right)+9\left(x+\...
Snowlash's user avatar
  • 115
25 votes
8 answers
1k views

Show that the polynomial $P(x):=x^4-6x+6$ has no real roots .

Show that the polynomial $$P(x):=x^4-6x+6$$ has no real roots. We need to solve this problem without using calculus. This is a problem from my son's olympiad textbook. Since the degree of the ...
hardmath's user avatar
  • 644
0 votes
1 answer
201 views

Determine the shape of a quartic polynomial given its equation only.

Some quartic (fourth-degree) polynomials resemble deformed parentheses, in that they only have one stationary point, which is their global maximum/minimum. This makes them share some properties of ...
GPWR's user avatar
  • 214
1 vote
6 answers
157 views

Solve $x^2+4y^2+80 = 15x+30y \\ xy=6 $

Solve for $x$ and $y$: $$x^2+4y^2+80 = 15x+30y \\ xy=6 $$ I have no idea how to solve this. I tried setting $x=\frac{6}{y}$ and plugging in, but all I get is this: $(\frac{6}{y})^2+4y^2+80 = 15(\frac{...
ronald christenkkson's user avatar
3 votes
3 answers
109 views

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 ...
J__n's user avatar
  • 1,123
5 votes
0 answers
86 views

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 ...
tituf's user avatar
  • 893
5 votes
1 answer
240 views

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 ...
Catherine's user avatar
3 votes
0 answers
138 views

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 ...
Akshat Saxena's user avatar
2 votes
3 answers
306 views

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 ...
Subeen Regmi's user avatar
0 votes
2 answers
119 views

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 ...
冥王 Hades's user avatar
  • 3,106
3 votes
6 answers
375 views

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 $x=\cos(t)$, $\sqrt{1-x^2}=t$, ...
rekophys's user avatar
1 vote
0 answers
77 views

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 ...
vvg's user avatar
  • 3,341
2 votes
0 answers
86 views

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 $y = 9u^2 + 7u + 2$, ...
Somebody's user avatar
1 vote
3 answers
207 views

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 ...
Vanessa's user avatar
  • 1,253
2 votes
2 answers
234 views

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 ...
user1114582's user avatar
11 votes
6 answers
772 views

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 ...
User's user avatar
  • 1,659
0 votes
0 answers
29 views

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)

I would like to transform the degree 4 polynomial to the degree 4 polynomial with specific powers of x removed. something like depressed cubic form or Brioschi quintic. Do we have any special ...
Ragavi's user avatar
  • 1
0 votes
1 answer
78 views

How to get roots of this 4th order polynomial given below?

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) ...
Gopalpur's user avatar

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