Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
1answer
46 views

If $a,b \in \Bbb [0,2]$ and $a+b=2$, maximize $ab(a^2+b^2)$ without differentiation.

If $a,b \in \Bbb [0,2]$ and $a+b=2$, maximize $ab(a^2+b^2)$ without differentiation. My try $a+b=2 \iff a=2-b \implies \max \{ab(a^2+b^2)\} \iff \max \{[(2-b)b][(2-b)+b^2)\}$ $\iff \max \{-2 b^4 + ...
0
votes
2answers
46 views

The set of values of $a$ for which $x^4+4x^3+ax^2+4x+1=0$ has real roots

The set of values of $a$ for which the equation $x^4+4x^3+ax^2+4x+1=0$ has real roots is given by $(-∞,m] U {n}$. Find the value of $√(n-m)$ I completed the square, and got the value of $m$ to be $6$,...
6
votes
3answers
148 views

Degree of splitting field of $X^4+2X^2+2$ over $\mathbf{Q}$

Find the degree of splitting field of $f=X^4+2X^2+2$ over $\mathbf{Q}$. By Eisenstein, $f$ is irreducible. By setting $Y=X^2$, we can solve for the roots: $Y=-1\pm i \iff X=\sqrt[4]{2}e^{a\pi i/8}$, $...
1
vote
1answer
109 views

Analytical solution to the crossed ladders problem

I'm working on an analytical solution to the crossed ladders problem. The solution is almost done and already useable (see my answer to Crossed Ladders Problem for details). However I'm left with a ...
0
votes
0answers
12 views

What kind of Planar Quartic Curve might this be?

I'm trying to smoke out the parameters for a family of curves showing up in a particular optimization problem. I have convinced myself that the solutions always lie on a quartic curve, which is ...
0
votes
2answers
66 views

Find the relation between $m$ and $n$ such that the following equation has four roots. [closed]

Find the relation between $m$ and $n$ such that the following equation has four roots with $m > 0$. $$x^2 + \left(\dfrac{mx}{m + x}\right)^2 = n$$ Well, I know what the answer is. I just want to ...
0
votes
0answers
52 views

interesting property of $x^4+2x^3+3x^2+4x+5$

I was recently looking at finding the minimum of the general quartic function $ax^4+bx^3+cx^2+dx+e$, for $a>0$. A closed form expression would, of course, be a huge mess, but it's easy to write ...
1
vote
0answers
29 views

Reduction Rule Approach Used in Factoring a Quartic

Can someone point me to an article or textbook where I can learn more about the reduction technique used in the third answer titled Rational Root Theorem Solution to the question asked in ...
0
votes
0answers
66 views

Factoring a general biquartic into two quartics

Let $W_0$, $W_1$, $W_2$, and $W_3$ be known real numbers. I have to solve a biquartic equation: \begin{equation} z^8+W_3z^6+W_2z^4+W_1z^2+W_0=0 \notag \end{equation} Of course I could solve the ...
7
votes
3answers
321 views

Factorise polynomial with real and complex roots

How would you go about finding the roots of the polynomial: $$x^4 +5x^3+4x^2+6x-4=0.$$ I attempt to form two quadratics e.g. $$(x^2+ax+b)(x^2+cx+d)$$ and then tried to expand, collect like terms, ...
1
vote
0answers
63 views

Name of the Quartic Surface $z=(x^2-a^2)(y^2-a^2)$

Does the surface defined by the following equation have an specific name? $$ f(x,y)=(x^2-a^2)(y^2-a^2)$$ I've searched a lot and found that $z=x^2y^2$ is called Crossed Trough. However I didn't find ...
1
vote
0answers
29 views

Multidimensional Quartic Equations

I know for the quadratic case (with $A$ an operator): $$ax^2 \Rightarrow x^T A x \Rightarrow \int xA[x]dx$$ Does any such analogy exist with $ax^4$ type functions? Either in the finite or infinite ...
0
votes
1answer
59 views

Solution of a fourth degree equation

Is there a viable strategy to solve the following equation in an analytic way, without using numeric methods? $(1+\frac{1}{8}x^2)^2=\frac{p^2}{2}(\sqrt{1+x^2}+1)$ Edit: When I substitute $t=\sqrt{1+...
1
vote
1answer
56 views

Solving for Ellipse Parameters Given a radius and angle (Challenge 2)

Given an ellipse centered on the origin in an x-y plane expressed as $$\bigg(\frac{x}{a} \bigg)^2+\bigg(\frac{y}{b} \bigg)^2 = 1$$ In polar coordinates with radius $R$ and angle = $\theta$, this can ...
0
votes
1answer
32 views

Estimating an equation from a series of values

Secondary school student here. I know we tend to get annoying on this site with fairly trivial questions and ignorance of rules, so feel free to educate me. What mathematical system could I research ...
0
votes
4answers
43 views

How to find the number of real roots for a polynomial?

How can you find of real roots for $P(x) = x^4 - 4x^3 + 4x^2 - 10$? Using the Descartes' rule of signs: The polynomial $P(x) = x^4 - 4x^3 + 4x^2 - 10$ has three sign changes between the first, ...
-5
votes
2answers
68 views

Apply Law of Exponents

Given $a^4+a^3+a^2+a+1=0,$ find the value of $a^{2016}+a^{2015}+1.$ Can anybody help me find the answer and teach me how to get it?
7
votes
4answers
186 views

How to prove $\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\ldots}}}}}}=\frac {\sqrt {4a-3}-1}2$

So, I was watching this video by blackpenredpen where he mentions that $$\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\sqrt {a-\sqrt {a+\ldots}}}}}}=\frac {\sqrt {4a-3}-1}2$$ so I wanted to try and prove it ...
0
votes
4answers
77 views

How can I find the real roots of $x^4-x^2+1$?

I've been trying to solve this for a couple of hours and I can't still find the answer. According to the answer I was given the reals roots should be: In reals: $\left(x^2+\sqrt{3}x+1\right)\left(x^2-\...
-1
votes
1answer
153 views

Problem on solving $z^4 - 2z + 4 = 0$ [closed]

I'm having a hard time solving this equation $z^4 - 2z +4 =0$. If you could help me how to solve it that would be amazing and also give me some references of how to solve equations like this ...
10
votes
3answers
872 views

Finding the sum of squares of roots of a quartic polynomial.

What is the sum of the squares of the roots of $ x^4 - 8x^3 + 16x^2 - 11x + 5 $ ? This question is from the 2nd qualifying round of last year's Who Wants to be a Mathematician high school competition ...
5
votes
3answers
90 views

Find value of $\prod(1+\alpha^2)$

if $\alpha,\beta,\gamma,\delta$ are the roots of equation $x^4+4x^3-6x^2+7x-9$ then find $\prod(1+\alpha^2)$ I know $\sum \alpha=-4$ $\sum \alpha\beta=-6$ $\sum \alpha\beta\gamma=-7$ $\alpha\beta\...
4
votes
5answers
109 views

Solve $8\sin x=\frac{\sqrt{3}}{\cos x}+\frac{1}{\sin x}$

Solve $$8\sin x=\dfrac{\sqrt{3}}{\cos x}+\frac{1}{\sin x}$$ My approach is as follow $8 \sin x-\frac{1}{\sin x}=\frac{\sqrt{3}}{\cos x}$ On squaring we get $64 \sin^2 x+\frac{1}{\sin^2 x}-16=\...
0
votes
0answers
98 views

Factoring quartic into 2 quadratic polynomials: $x^{4}+ax^{3}+bx^{2}+cx+d =(x^{2}+g_{1}x+h_{1})(x^{2}+g_{2}x+h_{2})$

I would like to factor the quartic into two quadratic polynomials $F(g),G(h)$: \begin{align*} x^{4}+ax^{3}+bx^{2}+cx+d & =(x^{2}+g_{1}x+h_{1})(x^{2}+g_{2}x+h_{2}),\\ & =x^{4}+(g_{1}+g_{2})x^{...
1
vote
2answers
207 views

How many zeros does the polynomial have in the right half plane?

The polynomial is $f(z) = z^4+\sqrt{2}z^3+2z^2-5z+2$ If you check the image of the imaginary axis, you see that there are no zeros, so we can use the right semicircle from $iR$ to $-iR$,and make $R$ ...
4
votes
1answer
113 views

Can we solve $A+D\sin^{2}x=B\sin x+C\cos x$ without having to solve a quartic polynomial?

Suppose the following equation $$ A+D\sin^{2}x=B\sin x+C\cos x, $$ where $A,B,C,D\in\mathbb{R}$ are the real constants. Initially, I tried to find its solution from a simple substitution \begin{align*}...
0
votes
0answers
46 views

Finding the minimum of a multivariate quartic function over a unit sphere.

Finding the minimum of a quadratic function over a unit sphere is simple I believe. You just have to find the minimum eigenvalue of the coefficient matrix. I would like to find the minimum of a ...
2
votes
2answers
377 views

Number of solution of $x^4-5x^3+(\lambda+2)x^2-5x+1=0$

Consider the bi-quadratic equation $E:x^4-5x^3+(\lambda+2)x^2-5x+1=0$ then, the real values of $\lambda$ so that $E$ has four different solutions is? My attempts: As $x=0$ is not a solution for any $...
10
votes
7answers
393 views

What are the steps involved in solving a quartic polynomial modulo a prime modulus?

This: $$x^4 + 21x^3 + 5x^2 + 7x + 1 \equiv 0 \mod 23$$ Leads to: $$x = 18 || x =19$$ I know this because of this WolframAlpha example and because a fellow member posted it in a since deleted & ...
2
votes
1answer
264 views

Really universal quartic formula

At the outset, I would like to say hello to the honorable discussants in this forum. [This is my first entry in this forum, so I apologize in advance for any possible mess.] I have a need to write ...
0
votes
2answers
42 views

nonlinear 3d system

I am trying to find all the critical points of my 3D Nonlinear System described as follows: \begin{array}{ll} \\ \dot{x}_1=\mu-x_1^2+x_3+x_2-2x_1 \quad (1)\\\\ \dot{x}_2=\mu-x_2^2+x_3+x_1-2x_2 \quad (...
0
votes
1answer
127 views

Finding range of rational function

$$y=\frac {x^2+ ax-2}{x-a}$$ I have been told that the range of the function is set of all real values then I am told to find the set of values of a. My attempt: $(x-a)y=x^2+ax-2$ $x^2+x(a-y)+ ay-...
1
vote
4answers
93 views

Can we say there is exactly one root of $x^4-7x^3+9=0$ in $(1, \:\: 2)$?

We have $f(1) \gt 0$ and $f(2) \lt 0$. Hence, the intermediate value theorem (IVT) guarantees at least one root in $(1, \: \: 2)$. Now, let's assume there are two roots $\alpha$ and $\beta$ in $(1, \:...
3
votes
1answer
81 views

Can't find an equation for calculating when 4 moving points have a circle passing through them

First thing - please forgive me if my way of explaining my problem is not formal or not accurate to standards, I am an amature mathematician and I have much to learn, I welcome you to let me know ...
3
votes
0answers
75 views

Integer solutions to diophantine equation $x^4+4x^3y-6x^2y^2-4xy^3+y^4=1$

While working on some research, I needed to find the integer solutions to the polynomial $x^4+4x^3y-6x^2y^2-4xy^3+y^4=1$. I found $(0,1)$, $(0,-1)$, $(1,0)$, $(-1,0)$, $(3,2)$, $(-3,-2)$, $(2,-3)$, $(...
0
votes
4answers
104 views

Arithmetical progression and quadratic equation

Determine the real number $k$ with the condition that the roots of the equation $x^ {4}-(3k+2) x^ {2} +k^ {2} =0$ make the arithmetic progression? I dont know how to start ?
1
vote
2answers
129 views

Median Age of Women at First Marriage ; Linear Algebra application problem

We are given an application problem: The median age of women in the United States at first marriage is given below. Let t= 0 correspond to 1970 and also let tbe measured in decades. Let 0 ≤ t≤ 4. ...
0
votes
0answers
50 views

How to get eigenvalue of 4 by 4 matrix out of each 2 by 2 matrices in quarters?

I have a very complicated 4 by 4 matrices with lots of complex variables in each entry. I can divide the matrix into quarters such as M = $\begin{pmatrix} M_1 & M_2 \\ M_2^\dagger & M_3 \end{...
1
vote
1answer
61 views

Prove that $x^4 + ax^3 + (b - 2)x^2 - ax + 1$ has $4$ real roots

The quadratic polynomial $x^2 + ax + b$ has exactly $2$ real roots. Prove that the quartic polynomial $x^4 + ax^3 + (b - 2)x^2 - ax + 1$ has $4$ real roots. I've tried Viete formulas and calculating ...
2
votes
2answers
1k views

Solving $x^4-15x^2-10x+ 24 = 0$ using Ferrari’s method

Ferrari’s method for solving a quartic equation $$x^4-15x^2-10x+ 24 = 0$$ begins by writing:$$x^4= 15x^2+ 10x-24$$and then adding a term of the form:$$-2bx^2+b^2$$to both sides. (a) ...
3
votes
2answers
91 views

How to solve a quartic equation with $x^4$ and $x$?

I am solving some heat transfer problems, and I came across this equation: $$(4.536 \cdot 10^{-8})x^4+ 12 x - 4316 = 0$$ The solution is $x = 320$ (I have the solutions book). I am using a HP50g ...
2
votes
4answers
525 views

Find the complex roots of this quartic polynomial

I have been trying to factor the polynomial $$x^4 - 2x^3 + 5x^2 - 5x +1$$ but the only root I can find is $(x-1) $. The context is that this is the characteristic polynomial of a matrix whose ...
0
votes
1answer
128 views

Quartic Formula existance [duplicate]

There is a formula for cubic equations and quadratic equations. Is there a formula for the equations in the form $ax^4+bx^3+cx^2+dx+f=0$ I also skipped $e$ because of Euler's constant.
0
votes
4answers
131 views

Find $y$ in $y^4-6y^3+14y^2-20y+8=0$

Find $y$ in $$\displaystyle y \cdot \frac{6-y}{y+1}\bigg(\frac{6-y}{y+1}+y\bigg) = 8$$ Solution I tried $y(6-y)(6+y^2)=8(y+1)^2$ $(6y-y^2)(6+y^2)=8(y^2+2y+1)$ $(36y+6y^3-6y^2-y^4)=8y^2+16y+8$ $y^...
0
votes
0answers
35 views

Largest rectangle that can be inscribed within a higher degree polynomial not broken up by the y axis?

So I understand how to solve for the largest rectangle that can be inscribed under a basic quadratic say $9-x^2$ (Using optimization), but I have no clue how to do it for an equation like $2x^4+4x^3+...
0
votes
1answer
87 views

Can I use Newton's method to obtain smallest positive real solution of a quartic?

I am familiar with Newton's method, but I'm not sure what convergence guarantees there are for this situation: I have a quartic in real coefficients $Ax^4 + Bx^3 + Cx^2 + Dx + E = 0$, and I need to ...
1
vote
1answer
35 views

To identify the types of roots [closed]

If $(x+2)(x+3)(x+8)(x+12)=4x^{2}$ then the equation has what type of roots? My attempt: Intersecting the graphs Can you please suggest any other easy method?
1
vote
2answers
92 views

Square inside a triangle problem [see desc.]

I Have a problem solving this problem only using high school math(Geometry) without using digital help (computer). The solution should be $\cfrac{11\sqrt{77}}{2}$. How can I get this solution by ...
2
votes
3answers
70 views

How to find the equation of this polynomial

How do you the simplified polynomial with integer coefficients and the following root(s): $\sqrt3+4i$ I think there is something related to conjugates, but I do not know how to do this. Due to the ...
1
vote
2answers
111 views

How to solve that?

I have no idea about how to solve the following: $$\sqrt[4]{13x+1} + \sqrt[4]{4x-1} = 3\sqrt[4]{x}$$ Could somebody help me, please?