Questions tagged [quartic-equations]

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.

Filter by
Sorted by
Tagged with
0
votes
3answers
167 views

Is there a quartic or quintic formula? [duplicate]

I know about the quadratic formula, and the cubic formula, so I was wondering if there were any more. My teacher said there was no such thing as a quintic formula, so I was wondering that if there was ...
0
votes
1answer
55 views

Solving $6x^4+2x^3+4x^2-6x-3=0$

I'm having some trouble solving for $x$ in the following quartic equation. $$ 0=6x^4+2x^3+4x^2-6x-3 $$ Do you have any suggestions on how I should go about solving this equation? I tried using the ...
1
vote
0answers
20 views

Lagrange quartic resolvent $x_1+ix_2-x_3-ix_4$

Suppose we want to solve the "reduced" quartic equation $x^4+px^2+qx+r=0$ by means of Lagrange resolvent. I denote the roots by roots $x_1, x_2, x_3, x_4$; we have $x_1+x_2+x_3+x_4=0$. In ...
3
votes
2answers
54 views

$f(f(x))=a^3\left(x^2-(2+b)x+2b-\frac2a\right)\left(x^2-(2+b)x+2b-\frac ba\right)$, $a\ne0$ has exactly one real zeroes $5$.

Let $f(x)=a(x-2)(x-b)$, where $a,b\in R$ and $a\ne0$. Also, $f(f(x))=a^3\left(x^2-(2+b)x+2b-\frac2a\right)\left(x^2-(2+b)x+2b-\frac ba\right)$, $a\ne0$ has exactly one real zeroes $5$. Find the minima ...
0
votes
1answer
70 views

Are these quartics polynomials having a real quadratic factor with complex conjugate roots?

I just want some help on some of these equations below to confirm whether they are polynomials having a real quadratic factor with complex conjugate roots? \begin{align} p(x)&=\frac{1}{3}(2x-6x+1)(...
2
votes
3answers
155 views

How to solve $x^4-2x^3-x^2+2x+1=0$?

How to solve $x^4-2x^3-x^2+2x+1=0$? Answer given is: $$\frac{1+\sqrt5}{2}$$ I tried solving it by taking common factors: $$x^3(x-2)-x(x-2)+1=0 $$ $$x(x-2)(x^2-1)+1=0 $$ $$(x+1)(x)(x-1)(x-2)+1=0$$ ...
3
votes
1answer
60 views

Solving complex quartic equation $z^5=1$

While finding the 5th roots of unity $z^5=1$, I arrived at the following $$(z-1)(z^4+z^3+z^2+z+1)=0$$ Now, I am well aware that I can arrive at the roots by using the fact that each root is separated ...
2
votes
4answers
94 views

Prove $X^4-2X^2+4$ is irreducible in $\mathbb{Q}[X]$

For some problem from my Galois Theory course, I need to prove that the polynomial $X^4-2X^2+4$ is irreducible in $\mathbb{Q}[X]$. I know it has no roots in $\mathbb{Q}$ (by rational root theorem), ...
0
votes
1answer
50 views

Proving that $x + y + z < 3abcd$

Suppose $a,b,c,d$ are real numbers greater than $1.$ Given that \begin{align*} a + b + c + d &= -x \\ ab + ac + ad + bc + bd + cd &= y \\ abc + abd + acd + bcd &= -z \\ abcd &= 858, \...
3
votes
2answers
172 views

Limit of Newton's Method on polynomial $Ax^4 + Bx^3+ Cx^2 + Dx + E$?

So if you took the function $f(x) = Ax^4 + Bx^3+ Cx^2 + Dx + E$ and did Newton's Method repeatedly, you would get a sequence that converges to at most $4$ roots. I was wondering what would happen if ...
8
votes
1answer
207 views

Factorization $x^4+px^3+qx^2+r x +s=(x^2+a x +b)(x^2+\bar a x +\bar b)$

Question: Under what condition, does the quartic polynomial with rational coefficients $p$, $q$, $r$ and $s$ factorizes as $$x^4+px^3+qx^2+r x +s= (x^2+a x +b)(x^2+\bar a x +\bar b) $$ with $a$, $b$ ...
0
votes
1answer
18 views

Complex quartic factorisation

Answer: I'm really not sure how to factorise. I understand that z.z* gives 2Re(z) but it's still not clear to me how it works.
1
vote
0answers
16 views

Diagonalizing 3-by-3 and 4-by-4 matrices using Givens rotations (solving 3rd and 4rth order polynomial equations)

The question is inspired by physics applications, where we are often interested in diagonalizing a Hamiltonian (a Hermitian matrix) by a unitary transformation: $$ S^\dagger H S = \Lambda, $$ where $H$...
2
votes
3answers
84 views

Factoring $x^4 + 12x^3 + 46x^2 + 59x + 18$

How do I factorize the following? $$x^4 + 12x^3 + 46x^2 + 59x + 18$$ I've tried looking for a root by trial and error to no avail. The answer is $$(x^2 + 5x + 2)(x^2 + 7x + 9)$$
-1
votes
2answers
51 views

Theory Of Equations : Prove that the roots are real [closed]

Prove that the roots of the equation $1/(x-1) +2/(x-2) +3/(x-3) =x$ ? Is real I have deduced by taking the recipocals and cross multiplying and its $4x^3-18x^2+10x+11 = 0$ not able to solve further. ...
0
votes
2answers
40 views

Determine a possible quartic polynomial equation such that $f(x) > 0$ for $-4 < x < -2$ and $3 < x < 7$

In typical high school fashion, nowhere in the curriculum was there a question about creating a polynomial equation. Yet here it is in the exam practice questions. Little help please?
0
votes
5answers
99 views

How to fully factorise $2x^4+7x^3+4x^2-4x$? [closed]

How to fully factorise $2x^4+7x^3+4x^2-4x$? I'm struggling to factorise polynomials like this one. I'm not sure how to best approach this problem. I've tried using the remainder and factor theorems ...
0
votes
1answer
20 views

Determining instantaneous rates of changes for quartic functions

I have 5 quartic functions that were found with quartic regression. Each function models a country's relationship with rotavirus vaccination rates (x) against years (y, years are integers and not real ...
0
votes
3answers
78 views

Solving a Quartic Function

A user named 'Uzdawi' from another post asked a question about how to solve the quartic function of One of the responses included an answer from the user 'Peđa Terzić', which is as follows: Could ...
0
votes
2answers
27 views

Quartic function sharing three common roots with another function

So the question is "The quartic function f(x) = (x^2+x-20)(x^2+x-2) has three roots in common with the function g(x) = f(x-k), where k is a constant. Find the two possible values of k." So ...
1
vote
1answer
52 views

Roots of a quartic form a geometric progression [closed]

Determine all real values of the parameter $a$ for which the equation $$16x^4 -ax^3 + (2a + 17)x^2 -ax + 16 = 0$$has exactly four distinct real roots that form a geometric progression.
5
votes
0answers
109 views

Who found the proof for the quartic formula?

A quartic equation is a 4th degree polynomial, in the form of $ax^4+bx^3+cx^2+dx+e$. There are 4 different formulae for the 4 roots of the quartic equation. Here are the formulae: $$x_1=-\frac{b}{4 a}-...
1
vote
2answers
43 views

Equation rearranging - what are $x$ and $y$ in terms of $u$ and $v$?

I am having trouble rearranging equations $u$ and $v$ into $x$ and $y$ in terms of $u$ and $v$. The equations are: $u = e^x - y$ and $v = y^2 +4e^{-2x}$. I wish to find $x = x(u,v)$ and $y = y(u,v)$ ...
1
vote
0answers
70 views

Automorphisms of Certain Quartic Equation

Let $B$ and $C$ be complex constants. Additionally, let $x,y,y,$ and $z$ be complex variables. What are the automorphisms of the quartic equation $((C-2)B-2C-12)(w^4+x^4+y^4+z^4)+((2(C-2))B^2+(2C^2+8C-...
0
votes
1answer
68 views

Rational Parameterization of Quartic Curve (Variety)

I am trying to find a rational parameterization of the curve (variety) $$x^4+a^2x^2=x^2y^2+(h^2+a^2)y^2$$ If I have done my math correctly, this curve has a singularity of multiplicity 2 at the origin,...
1
vote
3answers
92 views

Prove that $ f(f(x)) \geq 0$ for all real x

Let $f(x)= a x^2 + x +1 , x \in \mathbb{R} $. Find all values of parameter $a \in \mathbb{R} $ such that $f(f(x)) \geq 0 $ holds for all real $x$. $f(x)> 0 $ iff $a> 0 $ and $ 1- 4a \leq 0$ ...
1
vote
5answers
134 views

The equation $x^4-x^3-1=0$ has roots $\alpha,\beta,\gamma,\delta$. Find $\alpha^6+\beta^6+\gamma^6+\delta^6$

The equation $x^4-x^3-1=0$ has roots $\alpha,\beta,\gamma,\delta$. By using the substitution $y=x^3$, or by any other method, find the exact value of $\alpha^6+\beta^6+\gamma^6+\delta^6$ This is a ...
0
votes
1answer
42 views

Quartic Equation with a single variable

So, I got some tasks from my lecturer about quartic and even quintic equation Here is the question I tried using some general solution for quartic from this link Is there a general formula for solving ...
3
votes
2answers
84 views

Find value of $\sin x-\frac{1}{\cot x}$

If $\sin x+\frac{1}{\cot x}=3$, calculate the value of $\sin x-\frac{1}{\cot x}$ Please kindly help me Let $\sin x -\frac{1}{\cot x}=t$ Then, $$\sin x= \frac{3+t}{2}, \cot x= \frac{2}{3-t}$$ By ...
8
votes
3answers
120 views

Prove that $1 \leq A \leq \frac{5}{4}$ and $0 \leq B < \frac{81}{16}$

It’s known that A and B are real numbers it is also known that the polynomial P(x) has 4 real roots $$P (x) = x^4 − 3x^3 + 3x^2 − Ax + B$$ I did come up with a solution for A and I was hoping to apply ...
0
votes
2answers
120 views

Factoring question

Question: Factor $z^4 + 4z^2 + 6 - z.$ Here is the solution: Rewrite the given equation as $\left(z^2+2\right)^2 + 2 = z$. Observe that a solution to $z^2 + 2 = z$ is a solution of the quartic by ...
0
votes
5answers
111 views

How can I find the roots of the quartic polynomial $2x^4 −3x^3 +5x^2 +6x−4$?

I know that for a quartic polynomial $p(x)=ax^4+bx^3+cx^2+dx+e$ with $a=1$ one of the roots is a factor of $p(x)$. However here $a\neq1$, so I presume there is a trick to simplify this polynomial?
2
votes
5answers
166 views

How can I resolve this equation: $z^4 + 2z^3 + 7z^2 − 18z + 26 = 0$, where there is a root that it's $1+ i$

I know that if one root is $1+i$, other root is $1-i$. But I don't know how I can find the last $2$ roots. If you could explain in much detail why I always get lost in the little things, and sorry for ...
3
votes
2answers
108 views

What is the meaning of “due to the symmetry of the coefficients, if $x=r$ is a zero of $x^4+x^3+x^2+x+1$ then $x=\frac1r$ is also a zero”

I was studying this answer about factoring $x^4+x^3+x^2+x+1$: https://socratic.org/questions/how-do-you-factor-x-4-x-3-x-2-x-1 The author says: "A cleaner algebraic approach is to notice that due ...
5
votes
7answers
376 views

How can I prove that $p(x)=x^4+x+1$ doesn't have real roots?

Let $p(x)=x^4+x+1$ be a polynomial in $\mathbb{R}[x]$. How can I prove that $p$ doesn't have real roots? My attempt: From calculus, I know that $$\lim_{x \to \pm\infty} p(x) = \infty\,.$$ Then, if it ...
0
votes
4answers
172 views

Let $x_1,x_2,x_3,x_4$ denote the four roots of the equation $x^4 + kx^2 + 90x - 2009 = 0$. If $x_1x_2 = 49$, then find the value of $k$.

Let $x_1,x_2,x_3,x_4$ denote the four roots of the equation $$x^4 + kx^2 + 90x - 2009 = 0.$$ If $$x_1x_2 = 49,$$ then find the value of $k$. What I tried :- From Vieta's Formula for quartic equations ...
0
votes
2answers
50 views

need help solving a quartic equation

The question I am asking is to solve the equation $x^4-4x-1=0$, I need an exact answer. What I have done was found out that it equals $(x^2+1)^2 - 2(x+1)^2 =0$. Anybody help me, please?
3
votes
3answers
94 views

How can I find the roots of the polynomial $12x^{4}+2x^3+10x^2+2x-2$?

It's clear that I can divide by $2$, but I don't know what can I do with $$6x^{4}+x^3+5x^2+x-1$$ Is there any algorithm for it or a trick? I have found the roots by an online calculator but I don't ...
2
votes
3answers
103 views

Geometric sequence problem including sum of the numbers

Numbers: $a,b,c,d$ generate geometric sequence and $a+b+c+d=-40. $ Find these numbers if $a^2+b^2+c^2+d^2=3280$ I tried this problem and I have system of equations which I can't solve. I think there ...
2
votes
0answers
57 views

Beyond homography: from conics to quartics

It is known that a homographic transformation of a circle is a conic. The first figure is an example that fix the notation. Here the point $O$ is the center of the homography $h$. The axis of the ...
5
votes
4answers
183 views

Is there a quick (hopefully elementary) way to prove that $6b^2c^2 + 3c^2 - 36bc - 4b^4 - 4b^2 + 53=0$ has only one solution?

I have the Diophantine equation $$6b^2c^2 + 3c^2 - 36bc - 4b^4 - 4b^2 + 53=0.$$ Numerical calculations suggest this has only one positive integer solution, namely $(b,c)=(2,3)$. Is there a quick way ...
0
votes
2answers
56 views

Basic Geometry Problem : FInd $AC$

Let $ABC$ be a triangle where $\angle B$ is a right angle. Extend $AC$ up to point $D$ such that $\angle CBD=30^\circ$. If $AB=CD=1$, find $AC$. My approach to this problem is first to try to find ...
6
votes
8answers
407 views

Find all four roots of quartic equation $x^4-x+1=0$

How to solve $$x^4-x+1=0$$ My attempt: $$x^4-x+1=0$$ $$\implies x^4-x^3-x+1+x^3=0$$ $$\implies x^3(x-1)-(x-1)+x^3=0$$ $$\implies (x^3-1)(x-1)+x^3=0$$ But, I couldn't find a way to combine $x^3$ into ...
0
votes
3answers
64 views

Can someone help me solve this quartic equation?

$$x^4+5x^3-18x^2-10x+4=0$$ I cannot solve this quartic equation - is there any way to solve it apart from the quartic equation? It has no integer roots, and a hint given on the worksheet says to ...
1
vote
0answers
49 views

Minimization of Quartic Function

Hi I am dealing with an optimization problem with quartic function: $$x = \arg\min\limits_{x\in \mathbb{R}+} x^4 + (\frac{\alpha}{2} - 2y) x^2 - d\alpha x +y^2 + \frac{\alpha d^2}{2}$$ where $\alpha, ...
1
vote
1answer
49 views

I need help in solving this trigonometry related quartic equation.

I'm doing this calculation whereby I determine where the light source would be reflected on the sphere to the eye relative to its position. Assuming light travels in a straight line and the ray is ...
1
vote
1answer
114 views

Is there formula that solves quartic equation $ax^4+bx+c=0$

In general form, a quartic equation is $ax^4+bx^3+cx^2+dx+e=0$. I was thinking of the quartic equation of the form $$ax^4+bx+c=0$$ which resembles a depressed cubic equation. Does anyone know a ...
0
votes
1answer
226 views

How to find the equation of a quartic function given its three turning points

I am trying to find the equation of a quartic function given its three turning points. Here is the information I have: maximum turning points at $(-12, 1)$ and $(-6,1)$; minimum turning point at $(-9, ...
0
votes
2answers
86 views

let $a,b>0$,find the minimum of the value $f(a,b)=\frac{a}{b}+\frac{b}{a+b+1}+\frac{b+1}{a}$

let $a,b>0$,find the minimum of the value $$f(a,b)=\dfrac{a}{b}+\dfrac{b}{a+b+1}+\dfrac{b+1}{a}$$ I try $$f'_{a}=\dfrac{1}{b}-\dfrac{b}{(a+b+1)^2}-\dfrac{b+1}{a^2}=0$$ $$f'_{b}=-\dfrac{a}{b^2}+\...
2
votes
3answers
522 views

Find complex roots of quartic function $(3z + 1)(4z + 1)(6z + 1)(12z + 1) = 2$

I found a math problem involving complex number Find all complex number z such that $$(3z + 1)(4z + 1)(6z + 1)(12z + 1) = 2$$ The complex number form is z = a + bi If I multiply all the factor ...

1
2 3 4 5 6