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.

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41 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 ...
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43 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?
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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 ...
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3answers
83 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 ...
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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 ...
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4answers
161 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 ...
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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 ...
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8answers
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Find 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 ...
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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 ...
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Minimization of Quartic Function

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

A different kind of depressed quartic

In usual form, a quartic equation is $ax^4+bx^3+cx^2+dx+e=0$, for $[a,b,c,d,e]\in\mathbb{R}$. A depressed quartic is of the same form, but $b=d=0$ is the other rule. This simplifies down to a ...
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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, ...
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83 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}+\...
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74 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 ...
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Factorize $x^4 + x^3 - 3x^2 - 4x - 4$ showing all work

Factorize showing all workings $x^4 + x^3 - 3x^2 - 4x - 4$. I've attempted this question from the textbook "Core Maths for Advanced Level" by L. Bostock and S. Chandler and I'm having difficulty ...
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2answers
53 views

Show that $x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4=0$ has at least two solutions

Let $$p(x)= x^4 + a_1 x^3 + a_2 x^2 + a_3 x + a_4$$ be a polynomial with coefficients $a_1, a_2, a_3, a_4$. If $a_4 < 0$, then show $p(x)$ has at least 2 solutions. So far I thought, about using ...
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Problem using similarity of triangles (I guess).

I'm trying to answer the problem below using similarity of triangles, but wherever I go I got a fourth degree equation that is quite hard to solve. There's another way to solve this? I need a hint. ...
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1answer
59 views

Another way to solve the equation

Problem: Solve the equation: $$x^2+23x+23=(x+2)\sqrt{2(x^2+3x+6)}$$ My attempt After I squaring and simplifying, I got the following quartic equation: $$x^4-32x^3-531x^2-986x-481=0$$ This ...
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Number of real roots of $3x^4+6x^3+x^2+6x+3$

How many real roots does the following quartic polynomial have? $$3x^4+6x^3+x^2+6x+3$$ After dividing both sides by $x^2$, we get $$3x^2+6x+1+\dfrac6x+\dfrac3{x^2}=0$$ Or,$$3\left(x^2+\dfrac1{x^2}\...
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1answer
44 views

Quartic Formulation of tan(x) = x

I want to convert the above equation into quartic form. The only thing I can think of is to use the Taylor series expansion of tan(x). Is there any other way, anyone can suggest to convert the above ...
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3answers
134 views

The equation $x^4-2x^3-3x^2+4x-1=0$ has four distinct real roots $x_1,x_2,x_3,x_4$ such that $x_1<x_2<x_3<x_4$ and product of two roots is unity.

The equation $x^4-2x^3-3x^2+4x-1=0$ has four distinct real roots $x_1,x_2,x_3,x_4$ such that $x_1<x_2<x_3<x_4$ and product of two roots is unity, then: $Q-1$: Find $x_1\cdot x_2+x_1\cdot x_3+...
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71 views

Solving Quartic Equation with a Coefficient of $1$ MB Space

I have an equation of $4$ degree (Quartic equation)and a coefficient of this equation takes $1$ megabyte space in a text file. I want to solve this Quartic equation using computer. If the the equation ...
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How to use the quartic formula on this?

this is a follow up question to one I asked recently here: How to solve this quartic? I have the equation $x^4$+4a$x^3$+(4$a^2$+1)$x^2$−1=0, whereas I'm trying to find the solutions of x in terms of ...
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65 views

Is It Possible to Have Quartic Equation that Has Always Rational Root?

Let take the quartic equation $ax^4+bx^3+cx^2+dx+e=0$, how can we find/construct $a, b, c, d,e$ such that root $x$ is always a rational number? i.e. a general condition on $a, b, c, d, e$ such that ...
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69 views

Possible formula for $\sum_{} (\alpha\beta)^2$ and $\sum_{} \alpha\beta(\alpha+\beta)$

Given a quartic equation in the form $$ax^4+bx^3+cx^2+dx+e=0$$ and its roos $\alpha, \beta, \gamma$ and $\delta$, can anyone please help me find a formula for $$\sum_{} (\alpha\beta)^2 \quad \text{and}...
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84 views

How to solve this quartic?

I have the following equation, which I've put into the form of a normal quartic: $$x^4+4ax^3+(4a^2+1)x^2-1=0$$ I'm trying to find the solutions for $x$ in terms of $a$. Is there an easier way then ...
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74 views

Why does $3x^4 + 16x^3 + 20x^2 - 9x - 18$ = $(x+3)(x+2)(\frac{-1 \mp \sqrt 37 }{6})3$?

$$3x^4 + 16x^3 + 20x^2 - 9x - 18 $$ When simplified I arrive to: $$ (x+3)(x+2)(\frac{-1 \mp \sqrt 37 }{6}) $$ But the math book wrote: $$ (x+3)(x+2)(\frac{-1 \mp \sqrt 37 }{6})3 $$ with that extra ...
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solve the equation $ax^2-\sqrt(x)b-c=0$

I need to solve this system: \begin{align}y&=\sqrt x b+c\\ y&=ax^2\end{align} where the $x$ and the $y$ are the variable. Once I arrive to the equation: $ax^2-\sqrt x b-c=0$ I don't know ...
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72 views

Find all positive integers x such that $x^4-8x+16$ is a full square

Find all positive integers x such that $x^4-8x+16$ is a full square. I presented it as $(x-4)^2+x^4-x^2$ and found that $x = +1$ satisfy the condition. Then I equalized it $(x-4)^2+x^4-x^4=a^2+2ab+b^...
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Prove that this field is a splitting field

I was doing a proof to find the degree of the polynomial $X^4+9$, when I encountered a question. The roots of the polynomial in $\mathbb{C}$ are $$\pm\sqrt{6}\left(\frac{1\pm i}{2}\right)$$. So $\...
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1answer
75 views

Muller's method's initial approximations for non-real roots

What values should I use to obtain the non-real roots using Muller's method? The equation I am working on is: $$f(x) = x^4+2x^3+5x^2+5x-3$$ The equation has 2 real 2 non-real roots. I would like to ...
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Fourier transform of a quartic function

Fourier transform of inverse square root of a quadratic function is given by $$\int_{-\infty}^{+\infty}dy\frac{e^{i x y}}{\sqrt{y^2+a}}=2 K_0(\sqrt{a}|x|).$$ Is there any similar formula for a ...
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Number of real solution of Trigonometric Equation

Number of solution of the equation $(\sin x+\cos x+2)^{4}=128\sin(2x)\;\forall x\in\bigg[0,\frac{\pi}{2}\bigg]$ What i try $$\sin x+\cos x+2=\sqrt{2}\cos\bigg(x-\frac{\pi}{4}\bigg)+2$$ And put $\...
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A non-negative quartic function [duplicate]

Let $$f(x)=ax^4+bx^3+cx^2+dx+e$$ be a quartic function with $a,b,c,d,e\in\mathbb{R}$ and $a>0$. Could we find the necessary and sufficient condition on $a,b,c,d,e$ such that $f(x)\ge 0$ for all $...
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1answer
39 views

Expansion of the quartic of the sum of $N$ numbers

Can someone provide me the expansion for the following? $$A=\left( \sum_{n=1}^N a_n \right)^4 $$ I found square and cubic expansions here If there is any general expansion for power $k$ $$A=\left(...
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1answer
76 views

a quartic polynomial in one variable

Let $$g(x)=ax^4+bx^3+cx^2+dx+e$$ be a polynomial of degree $4$ with $a>0$. Can we determine coefficients $a,b,c,d,e$ such that $g(x) \ge 0$ for all $x\in\mathbb{R}$? I'd like to make sure that ...
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50 views

Should I put any coefficient in front of $x^4$, or does it matter?

I was trying to solve the problem: Given that a and b are distinct positive numbers, find a polynomial P(x) such that the derivatives of $P(x)e^{-x^2}$ is zero for $x=0, x=a, x=-a, x=b, x=-b$. We ...
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63 views

How to determine number of roots and what type for quartic equations?

I want to figure out the number of roots and their types in the quartic equation $$x^4-34x^2-x+272=0$$ without actually solving it. Is there such a way to do so?
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Why are the various invariants of the quartic not symmetric?

To find if a quartic has real roots, one has to look at various quantities made from the coefficients. $$ax^4+bx^3+cx^2+dx+e=0$$ But I would've expected these quantities to be invariant under the ...
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Solve this “quartic” guassian integral.

What is the solution to this integral (for when it converges)? $$\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty e^{-ax^4-bx^3y-cx^2y^2-dxy^3-ey^4} dx dy$$ Free variables are $a,b,c,d,e$.
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Is it always possible to solve a quartic equation with real coefficients without operating with complex numbers?

I'm working on an algorithm to calculate all real roots of a quartic equation. At the moment I have a solution using the method of Descartes and Vietas substitution. My approach is: Normalize ...
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1answer
39 views

Use resolvent (or anything else) to find Galois group of quartic equations

Like I mentioned in the question, I am trying to find a way to find the Galois group for a general quartic polynomial. I am reading the book Galois Theory of Escofier. He solved the case of cubic ...
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Galois group of $x^4+7x+7$

How can I compute the Galois group of $x^4+7x+7$ over $\mathbb{Q}$? I believe it can be done using some general results about discriminants and cubic resolvants such as in this document: https://...
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1answer
31 views

How to find the area between quartic functions?

Assume I have following two equations: $f(x) = a_1 x^4 + b_1 x^3 + c_1 x^2 + d_1 x + e_1$ $g(x) = a_2 x^4 + b_2 x^3 + c_2 x^2 + d_2 x + e_2$ Can I calculate $\int_{-5}^5 |f(x) - g(x)| dx$ using ...
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1answer
46 views

Family of quartics given the only two roots and an extremum

How to find the family of quartics with only the two roots $2$, $10$, and one extremum $(-5,5)$? It should look like this: I want a solution with "roots-factors", like this: $$f(x) = a((x-x_A)(x-...
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165 views

What can be a real world application for solving quartic equations?

I have been learning the possible solutions for this type of equations but I have no idea when and how it can be used in real life. Do we have any problematic example for a real life application of ...
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1answer
34 views

Polynomial (quartic) with complex coefficient: root localization

For all $u\in \mathbb C$, let $P_u(X)= X^4+4X+u$. I know that the roots of $P_u$ could be found explicitly, but it seems to lead to inextricably complex calculations. Let $H= \{z \in \mathbb C ~|~ \...
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1answer
38 views

Stabilizer of binary quartic

While studying a larger problem I came across the question of which binary quartic forms have finite stabilizer. More precisely. Let's denote five-dimensional irreducible representations of $SL_2(\...
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2answers
124 views

How to find roots of a 4th order polynomial?

Given that $y=14+z$ and $y=z^4$ find the value of $y$. Substituting $y=z^4$ we get: $z^4 = 14 + z$ $\Rightarrow$ $z^4 - z - 14 = 0 $ I do not know how to approach solving this polynomial. The ...

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