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.
340
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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 ...
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0
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84
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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 ...
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1
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1
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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
$$\mathcal{E}=\{x\mid (x-x_c)^TP^{-1}(x-...
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2
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42
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Coefficient of a quartic polynomial given the conditions it's a natural number and has atleast one real root
Let $y=f(x)$ be a curve given by the solution of differential equation $dy/dx=4px^3-12x-8$ , $p ∈ N$, $f(0) = 24$ and $y=f(x)$ cuts x-axis at atleast one point. Find number of value(s) of p and sum of ...
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4
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Find $\sin{\frac{\pi}{10}}$ in form of $a+b\sqrt{k}$
Use De Moivre's Theorem to express $\cos{5\theta}, \sin{5\theta}$ in powers of $\sin{\theta}$ and $\cos{\theta}$.
Show that $$\frac{\cos{5\theta}}{\cos{\theta}}=16\sin^4{\theta}-12\sin^2{\theta}+1$$
...
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2
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Show that the roots of the equation $b^2(x^4+1)-14b(1+b^2)x(x^2+1)+(1+196b^2+b^4)x^2=0$ are all real and all have the same sign as $b$.
If $\alpha, \beta$ are the roots of the equation $$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 $...
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2
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2
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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 $...
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1
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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) ...
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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{...
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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 ...
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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 ...
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Root representation using the factorization of polynomials upto degree six
Is there some classical technique (from algebra or analysis) to find the expressions of roots of quintic and sextic using the root expressions of quadratic, cubic and quartic root expressions.
For ...
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1
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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\...
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8
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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:
$$ z^4 + z^3 + z^2 +2z +3 = \left( z^2 + \frac12 z - \frac18 \right)^2 + \left( z + \frac{17}{16} \right)^...
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Can the local minima and maxima of a quartic be at any (unique) arbitrary point?
Is it possible to construct a quartic function with two local maxima and one local minima at any three arbitrary points, making some assumptions? If so, how can I do it?
The x-coordinate of the local ...
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investigated by descartes rule of sign
Use Descartes' rules of signs to discuss and determine (how many) the number of possible positive roots of each equation.
$p(x)=ax^4+bx^3+cx^2+dx+e=0$
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2
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Find the sum of the real roots of the polynomial $P(x)=(x^2+2x+4)^2+2(x^3-8)+(x-2)^2$
Find the sum of the real roots of the polynomial
$$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,$$ I think ...
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How to factor quartic into quadratic factors? [duplicate]
I am having problems factoring the following quartic with quadratic roots.
$$3x^4 + x^3 + 2x^2 - x + 1 = \left(x^2 + x + 1 \right) \left( 3x^2 - 2x + 1 \right)$$
All the ways I've found online till ...
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Find the sum of the real roots of the equation $6x^4+9x^3-15x^2+9x+6=0$
Find the sum of the real roots of the equation $$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+\...
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8
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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 ...
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1
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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 ...
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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{...
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3
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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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6
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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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0
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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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2
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215
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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)
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 ...
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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)
...
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2
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Show that the polynomial $P(x)=x^4-x^2-x+2$ has no real roots [closed]
Using clever algebra show that the polynomial
$$P(x)=x^4-x^2-x+2$$
has no real roots.
Obviously, we can not use the derivative.
Using the general quartic formula is terrible.
I tried
$$(x^2+1)^2-3x^...
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1
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How to solve square of Quadratic Equation
The formula looks like this:
$$
\bigl( a x^2 + b x + c \bigr)^2 = \; d
$$
What I try to solve is the following system:
\begin{align}
d_1 &= (a_1 x^2 + b_1 x + c_1)^2 \tag{1} \\
d_2 &= (a_2 x^2 ...
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1
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4
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The equation $x^{4}-3x^{3}-6x^{2}+ax+b=0$ has a triple root. Find $a$ and $b$, and hence all roots of this equation.
The given question is:
The equation $x^{4}-3x^{3}-6x^{2}+ax+b=0$ has a triple root. Find $a$ and $b$, and hence all roots of this equation.
I am confused about how to work out this question, but I ...
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2
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103
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is there a cubic or quartic approximation for these data points? [closed]
Is there a cubic or quartic approximation for these data points; $$(0,1000000), (1000000, 100), (10000000, 10)$$ whilst also ensuring that no point from $$1000000 < x < 10000000$$ is greater ...
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3
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Prove $f(-\frac12) \le \frac{3}{16}$ if all roots of $f(x) = x^4 - x^3 + a x + b$ are real
Let $a, b$ be real numbers such that all roots of $f(x) := x^4 - x^3 + ax + b$ are real. Prove that $f(-1/2) \le 3/16$.
The question was posted recently which was closed,
due to missing of contexts ...
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Square roots of root-expressions: distinguishing quartic Galois groups from copies of $C_4$ or of $D_4$ - exercise M.11, Ch.16 Artin's algebra
$\newcommand{\gal}{\operatorname{Gal}}$The setup:
$F$ is some field (Artin always assumes characteristic zero to get separability, but I don't like this choice) and $f(x)\in F[x]$ is an irreducible ...
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1
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51
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Resting a stick in a cup. How far away from the side will it be?
So this is a question someone else asked me, and I haven't had a chance to figure out if it might be an X/Y problem, but for now let's assume not.
We have a (rectangular crossection) cup of height $v$ ...
- 5,157
0
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0
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84
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Quartic with only one singular point
I'm trying to proof that the Curve $C:xz^3-y^4+2y^2z^2-z^4=0$ is irreducible for all fields K, with $char(K)>2$.
I know that the point $(1:0:0)$ is the only singular point and its multiplicity is 3....
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1
answer
124
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What is the possible sum of non real roots$?$ [duplicate]
Consider an equation $$x^4-2x^3+2x^2-x+k=0$$ For every real number $k$
Find the minimum number of non real roots.
The sum of non real roots can be___
My work:
For the first part I got the right ...
user1074471
3
votes
2
answers
136
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Is there a "magic" binary operation from which addition and multiplication can be derived?
I planned to make a one instruction set computer (abbr. OISC), and this question arose.
I wanted a "magic" binary operation $*: \mathbb{Z} × \mathbb{Z} → \mathbb{Z}$ so addition and ...
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3
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497
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How to get the general term for a quartic sequence really need help [closed]
Is it possible to find the general term for a quartic sequence and if so how do you do it?
The sequence I am using is 1,9,36,100,225,441, 784, 1296, 2025, 3025
I am only interested in finding the ...
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1
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0
answers
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Is it true that when a root of a 4th degree polynomial is natural, the radicals inside the formula are always rational?
Given the formula for the 4th degree polynomial, is it true that a root is a natural only when all the radicals inside the formula are rational numbers?
Edit1: The coeficients are whole numbers.
https:...
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1
answer
128
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Mean of Cubic and Quartic forms of Gaussians
I am trying to calculate the following means:
$$ E[ (x-\mu_k)b^T(x-\mu_l)(x-\mu_l)^T ] $$
$$ E[ (x-\mu_k)(x-\mu_k)^TA(x-\mu_l)(x-\mu_l)^T ] $$
Where x is some multivariate gaussian random variable.
I ...
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1
vote
1
answer
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System of Two equations with two unknowns of degree four
I am wondering if there is a direct way to solve exactly a system of two equations of this shape (the A to I are constants):
$Axy + Bxy^2 + Cx^2y + Dx^2y^2 + Ex^2 + Fy^2 + Gx + Hy + I=0$
this problem ...
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