Questions tagged [cubics]

This tag is for questions relating to cubic equations, these are polynomials with $~3^{rd}~$ power terms as the highest order terms.

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Is $a+5^{1/3}b+5^{2/3}c$ a root of any cubic polynomial in $\mathbb{Q}$?

For arbitrary $a, b, c \in \mathbb{Q}$, let $w := a + 5^{1/3} b + 5^{2/3} c$, is $w$ a root of any cubic polynomial in $\mathbb{Q}$? I guess the cubic polynomial always exists. But I am confused about ...
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1 vote
0 answers
36 views

How to find $x$ by $y$ in bezier curve.

I draw the lines connecting the different nodes on the screen with the cubic bezier curve. These lines must be suitable for interaction. So I need to know how close the user is to the line based on ...
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0 votes
0 answers
47 views

Meaning of $ g(x) = f(x-1) \times \left|\lim_{ h\to 0} \frac{f(x+h)-f(x)}{h}\right| $ has only one non-differentiable point?

$f(x)$ is a cubic function with leading coefficient of 1, $$ g(x) = f(x-1) \times \left|\lim_{ h\to 0} \frac{f(x+h)-f(x)}{h}\right| $$ $g(x)$ is not differentiable only at $x= -1$. $g(1)=0$, $f'(0)&...
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How to deal with functions like $|f(x) + x| = px$ when $f(x)$ is a cubic?

The problem is : There is a positive real number $p$, and a cubic function $f(x)$ with its leading coefficient $1$. Every real roots of an equation $f(x) = px$ is $\alpha - \beta$, $\alpha$, $2\beta$. ...
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  • 45
-1 votes
2 answers
26 views

finding the equation of a cubic function with 2 points [closed]

A cubic function is in the form $y=kx^3+c$. Find the equation if it passes through $(0,5)$ and $(2,-3)$. I’m not sure how i would work this out algebraically or if i even can without graphing.
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1 vote
1 answer
60 views

What can be derived from the fact that $\lim_{x\to 0}\frac{f(x-3)f(x+3)}{x^3}$ is convergence?

Problem is : There is a cubic function $f(x)$ , with its positive coefficient. And, $\lim_{x\to 0} {\frac{f(x-3)f(x+3)}{x^3}}$ is convergence. there is only one natural number $k = k_1$ which makes ...
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0 votes
1 answer
63 views

Polynomial and Complex Roots Problem

Suppose that there exist nonzero complex numbers $a,$ $b,$ $c,$ and $d$ such that $k$ is a root of both the equations $ax^3 + bx^2 + cx + d = 0$ and $bx^3 + cx^2 + dx + a = 0.$ find all possible ...
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2 votes
3 answers
113 views

Solving $a^3 - 33 ab^2 = -217$ and $3a^2 b - 11b^3 = 18$?

I am looking for ways of solving systems like that: $$\left\{\begin{array}{lcl} a^3 - 33 ab^2 = -217 \\ 3a^2 b - 11b^3 = 18 \end{array} \right.$$ I've tried turning it into a system of 2 equations ...
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4 votes
2 answers
102 views

Systems of cubic equations [closed]

I am looking for a general yet tailor-made methods to solve system of equations like that $a^3 + 6ab^2 = 7$ and $3a^2b +2b^3 = 5$ that involve terms of $(a +b)^3$. Does this have a name?
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4 votes
0 answers
81 views

A cute way to solve the quadratic. How to extend it to the cubic?

Playing around with complex numbers, I found a cute way of solving the quadratic equation. Let's start with the (monic) equation \begin{equation} z^2+pz+q = 0 \end{equation} where $z$ and the ...
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0 votes
1 answer
55 views

Why doesn't the Cardano Formula yield the result expected by the inverse equation?

My application is about indoor bicycles. Sometimes I need to estimate the virtual speed for certain amount of Watts on the pedal or the other way around (how much watts would a certain speed require ...
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0 votes
3 answers
92 views

Constructing a tangent line to a point on the curve $y = x^3$ using a compass and straight edge?

There are number of ways of constructing a tangent line to the curve $y = x^2$ using a compass and straight edge. Does anyone know of a way of constructing a tangent line to the curve $y = x^3$ using ...
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0 votes
0 answers
42 views

Determining the roots of the cubic polynomial

Let's say we have a third-order polynomial given as follows: $P(u)=-u^{3}+a_{1}u^{2}+a_{2}u-a_{3}^{2}$, where $a_{1},a_{2},a_{3}$ are all constants. In the specific problem that I am dealing with it ...
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15 votes
1 answer
202 views

Does the limit of the cubic formula approach the quadratic one as the cubic coefficient goes to $0$?

The formula for solving a cubic equation of the form $ax^3+bx^2+cx+d=0$ does not seem to yield the quadratic formula for the limit $\lim _{a \rightarrow 0} \text{(cubic formula)}$. But, if one tries ...
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  • 197
1 vote
2 answers
78 views

Solving the system $a^3 + 15ab^2 = 9$, $\;\frac 35 a^2b + b^3 = \frac 45$

I have problem solving the following system of two cubic equations. $$\begin{cases} a^3 + 15ab^2 = 9 \\ \frac 35 a^2b + b^3 = \frac 45 \end{cases} $$ I don't have any idea how to approach this kind of ...
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2 votes
1 answer
68 views

Discriminant of depressed cubic [duplicate]

If the cubic equation $x^3+px+q$ has roots $\alpha , \beta , \gamma $ then we know that $\alpha + \beta + \gamma =0 $, $\alpha \beta + \alpha \gamma + \beta \gamma =p $ and $\alpha \beta \gamma =-q $. ...
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4 votes
1 answer
102 views

Solving a depressed cubic polynomial in modulus. [closed]

Is there a general technique for solving depressed cubic modulus polynomial? For instance, how would you solve the equation $a^3 + a + 21 = 0 \pmod{43}$?. My attempts eventually ended up with solving $...
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0 votes
2 answers
97 views

Factoring $x^3-6x^2+11x-6$ without using Rational Roots Theorem

To factor $P(x)=x^3-6x^2+11x-6$ one way is using Rational Roots Theorem and recognizing that $x=1$ makes $P(x)$ zero. But I want to factor without using it. I tried, $$x^3-6x^2+11x-6=x^2(x-6)+11(x-6)+...
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  • 1,007
0 votes
0 answers
73 views

Is this identity $ (1+6x^3+9x^4)^3+(1-6x^3+3x-9x^4)^3+(1-9x^3-6x^2)^3=9x^2+9x+3$ known with $x$ is an integer?

It is known that, let $x$ arbitrary integer $$(9x^4)^3+(3x-9x^4)^3+(1-9x^3)^3=1\tag{1}$$ discovred by Kurt Mahler in 1936, and $$(1+6x^3)^3+(1-6x^3)^3+(-6x^2)^3=2\tag{2}$$ discovred by A. S. ...
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3 votes
2 answers
58 views

Minimum value of M such that the cubic modulus value is always less than M for x in between -1 and 1 both included

Minimum value of $M$ such that $\exists a, b, c \in \mathbb{R}$ and $$ \left|4 x^{3}+a x^{2}+b x+c\right| \leq M ,\quad \forall|x| \leq 1 $$ What i considered was that putting x= 0, 1 and -1 we get ...
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  • 381
0 votes
1 answer
34 views

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
0 answers
16 views

What the extra 3 roots are of the 6th degree equation ? From which only the intial 3 are what we required in the problem

Question was to find the roots whose roots are square of the given roots of this polynomial $x^{3}+a x^{2}+b x+c=0$ I did used tranformation method to obtain $y^{3 / 2}+a y+b y^{\frac{1}{2}}+c=0$ but ...
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0 votes
2 answers
30 views

Finding equation whose roots are square of the original but intially some of them were negative .

Consider a cubic given as $x^3 + ax^2 + bx + c = 0$ which has roots $\alpha,-\beta,-\gamma$ , we need to find cubic equation whose roots are $\alpha^2 ,\beta^2 , \gamma^2$ is it possible to get it by ...
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1 vote
1 answer
33 views

Main Motivation behind the use of identity of $(a+b)^3$ ,so as to solve cubic equations using Cardons Method [closed]

Whats so special about the identity $(a+b)^3 = a^3 + b^3 + 3ab(a+b)$ which gave Cardano the clever idea to invent a method to solve cubic depressed equations?
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  • 381
0 votes
1 answer
32 views

How to get the coefficients in a parametric cubic function

Let's say I have 4 points with x and y coordinates. And I want to determine the parametric cubic function: $x(t) = a_x t^3 + b_x t^2 + c_x t^3 + d_x$ and $y(t) = a_y t^3 + b_y t^2 + c_y t^3 + d_y$ So, ...
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0 votes
0 answers
25 views

2 Other Inverse Function of $f(x) = x^3+ax, \; a < 0$ for $x^2 \leq -\dfrac{4a^3}{27}$

For $a > 0$ the unique inverse of $f(x) = x^3+ax$ is $f^{-1}(x) = \dfrac{\sqrt[3]{9x+\sqrt{81x^2+12a^3}}+\sqrt[3]{9x-\sqrt{81x^2+12a^3}}}{\sqrt[3]{18}}$ But for $a < 0$ i know that this isn't ...
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0 votes
3 answers
112 views

Find the roots of $z^3 +3z^2 +3z +3=0$

Hello I have this problem: Find all $z \in C$ $z^3 +3z^2 +3z +3=0.$ With Mathematica I get 3 roots $z_1 = -1 -\sqrt[3]{2}$ $z_2 = -1 + (1 + i \sqrt{3})/2^{(2/3)}$ $z_3 = -1 + (1 - i \sqrt{3})/2^{(2/3)}...
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0 votes
1 answer
79 views

Inverse a cubic function

I've got this cubic function that I can't figure out how to calculate its inverse. $$f(x)=x^3+3x^2+3x, x\in \mathbb{R}$$ I've tried using online calculator to see if that would help, but none of them ...
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  • 25
0 votes
2 answers
35 views

Find a cubic function given the inflection point and minimal point

I came across this task that I just couldn't figure out. The task gave me an extremum(1,1) and the inflection point(2,3), and I need to figure out the cubic function given the values. Assuming the ...
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  • 25
0 votes
1 answer
30 views

How to factor $z^3-3\sqrt{3}iz^2-9z+3\sqrt{3}i$ strictly via the method of grouping

I really don't have much of my own attempt to show for this question. All I managed to get to was: $$z^2(z-3\sqrt{3}i)-3(3z-3\sqrt{3}i)$$ By simply factorising. It is asked in the question to be ...
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3 votes
2 answers
78 views

Question on cubic polynomials and roots

Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roots of $f(x)$ are $r + 1$ and $r + 7.$ Two of the roots of $g(x)$ are $r + 3$ and $r + 9,$ and$$f(x) - ...
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0 votes
3 answers
106 views

Cubic polynomial without $x^1$

I factorized this equation $4x^3+6x^2-1=0$ I think it's harder to factorize when the $x$ (degree one) doesn't exist here. So, this is what I did now: $2x^2(2x+3)-1=0$ But the factorized equation looks ...
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0 votes
2 answers
82 views

solve this formula for $x$: $y = \frac{5x(2x^2 + 27x + 89)}{6}$

I need to solve this formula for x: $$y = \frac{5x(2x^2+27x+89)}{6}$$ When inserting this into a Formula calculator it gave me the formula in the attached image, however, I have no clue how to use ...
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2 votes
3 answers
70 views

Stuck on simplifying expressions involving trig and inverse trig functions

TL; DR Using Mathcad and Wolfram I can see that $$\sqrt{7}\cos\frac{\tan^{-1}\left(\frac{9\sqrt{3}}{10}\right)}{3}=2.5$$ The decimal value seems to be exact because Mathcad displays it like that with ...
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  • 163
0 votes
1 answer
57 views

What are the necessary and sufficient conditions for the cubic equation to have at least 1 positive real root?

What are the necessary and sufficient conditions for the cubic equation to have at least 1 positive real root? I'm just dealing with the $2$ simplest cases. Case-1: $$x^3+px+q=0$$ where, $p<0$ and ...
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  • 597
2 votes
2 answers
84 views

Condition for real roots of depressed cubic equation

My book sets out to prove that three normals drawn to any parabola $y^2=4ax$ from a given point $(h,k)$ are real when $h>2a$. It gives the following proof: Proof: When normals are real, then all ...
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2 votes
3 answers
94 views

Find $a, b, c$ such that $a^3+b^3+c^3-3abc=2017$.

Find all natural numbers $a, b, c$ such that $a\leq b\leq c$ and $a^3+b^3+c^3-3abc=2017$. My Attempt $$a^3+b^3+c^3-3abc=2017$$ $$(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=2017*1$$ Now, $a+b+c$ can't be equal to $...
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1 vote
1 answer
75 views

Alternative Solution for Cubic Equation

I was solving a problem and got to the following equation $x^3 + x^2 + x - 1 = 0 \; \; (1)$, numerically I found that the solution was: $$x =\frac{1}{3} \left(-1 - \frac{2}{\left(17 + 3 \sqrt{33} \...
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  • 336
7 votes
2 answers
120 views

Clarification about solving cubic equations

I'm trying to understand a statement written on Wikipedia about solving cubic equations. In particular this part: A ''cubic formula'' for the roots of the general cubic equation (with $a\neq 0$) $$ax^...
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0 votes
1 answer
61 views

A perplexity over the cubic root.

Good morning; I'm having doubts about a perhaps simple question involving the cubic root of a function. Say I do have the following $$f(x) = \sqrt[3]{\dfrac{x^3}{2-x^2}}$$ Its domain is clearly $x\neq\...
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2 votes
1 answer
76 views

Understanding Cardano's Formula

In deriving his formula, Cardano arrives at the equation $y^3+py+q=0$. By substituting $y=\sqrt[3]{u}+\sqrt[3]{v}$, he gets the equation $(u+v+q)+(\sqrt[3]{u}\sqrt[3]{v})(3\sqrt[3]{u} \sqrt[3]{v} +p)=...
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3 votes
3 answers
181 views

If $\alpha,\beta,\gamma$ are the roots of $x^3+x^2-x+1=0$, find the value of $\prod\left(\frac1{\alpha^3}+\frac1{\beta^3}-\frac1{\gamma^3}\right)$

If $\alpha,\beta,\gamma$ are the roots of the equation $x^3+x^2-x+1=0$, find the value of $\prod\left(\frac1{\alpha^3}+\frac1{\beta^3}-\frac1{\gamma^3}\right)$ My Attempt: $\alpha+\beta+\gamma=-1$ $\...
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  • 5,384
2 votes
2 answers
68 views

Asymptotic expansion of a cubic equation

I am asked to find the first two terms of the asymptotic expansion of the cubic equation $$\varepsilon^4 x^3 - 6 \varepsilon^3 x^2 + (4-3\varepsilon^2) x - 12 \varepsilon - 2 \varepsilon^2 = 0$$ as $\...
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  • 21
0 votes
2 answers
58 views

Why do the 3 solutions of $x^3 + ax^2 + bx + c$ add up to $-a$?

Elliptic Tales makes the claim that: The three solutions of the equation $x^3 + ax^2 + bx +c = 0$ add up to $-a$. The reason is: Why? Suppose that the three solutions are $x_1,x_2,x_3$. Then we ...
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  • 129
0 votes
0 answers
48 views

Expressing the solution to a rational expression in radical form

I'm trying to find the solution to this equation $$ -\frac{3}{r} + \frac{8}{r^3} = \frac{\sqrt{2}-1}{2} $$ but I haven't been able to find a solution in radical form. Although I've found the solutions ...
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  • 1
0 votes
0 answers
116 views

The Diophantine equation $x^2\pm iy^2=z^3$ in Gaussian integers

I am trying to find any reference about the solutions of the Diophantine equation $x^2\pm iy^2=z^3$. It seems to me that I read about this somewhere before, but I can't remember where. Any ...
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0 votes
1 answer
53 views

Trajectory of a 2D constant jerk motion

This thread is a natural prosecution of this other. Problem Let $\xi,\eta$ be the horizontal and vertical coordinates of a plane. Consider the following sequence of point \begin{equation}\begin{...
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0 votes
0 answers
62 views

Finding the roots of an absolute value natural log function, regarding the usage of Newton-Raphson

This is an exercise I saw in my last test, I've been practicing it but I'm currently a bit lost when finding the roots, here's the exercise: Given $f:f(x)=-x^2+2x+3$, be F/F(x) is a primitive of f ...
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  • 31
2 votes
1 answer
193 views

I'm failing to solve a cubic equation

SOLUTION: Because I made the substition of $t = x + \frac {b}{3a}$, I had to substract $\frac {b}{3a}$ from my final answer, giving me the formula for x: $x = u + v - \frac {b}{3a}$. I'm trying to ...
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2 votes
3 answers
125 views

relation between roots and coefficient in a cubic polynomial

If $\alpha,\beta,\gamma$ are roots of the cubic equation $$2x^{3}+3x^2-x-1=0$$ then I want to find the equation whose roots are $\frac{\alpha}{\beta+\gamma}, \frac{\beta}{\gamma+\alpha}, \frac{\gamma}{...
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