Questions tagged [cubic-equations]

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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9 views

when do Lagrange resolvents work to solve a polynomial such as the following:

y^3 - 3y^2 - 3y + 1 = 0 to this one it works but when i try it on the one below : y^3 - 2y^2 + 1 = 0 the Jagrange resolvent technique does not work, I would like to know why.
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Analytically solving $\frac{1}{\sin2x} + \frac{1}{\sin3x} = \frac{1}{\sin x}$

Given $$ \frac{1}{\sin(2x)} + \frac{1}{\sin(3x)} = \frac{1}{\sin x}$$ I tried solving the equation above using the double and triple angle formulas and arrived at this cubic expression in $\cos x$ $...
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21 views

Optimization of Quadratic Equation with Wirtinger Calculus

The problem is to find the complex-valued solution $z$ of the real-valued cost function $f(z)$: $$z^* = \mathop{argmin}_\limits{z\in \mathbb{C}} (f- |z|^2)^2 + \alpha|z-d|^2,$$ where $f, \alpha \in \...
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18 views

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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28 views

$x^3-bx^2+4hx-ph=0$

$$x^3-bx^2+4hx-ph=0$$ Taking the equation above, is there away to prove that for every positive prime p: There are positive integer values for b and h so that the cubic has 3 positive integer roots ...
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2answers
34 views

Is there a rule for evaluating a sum like $\sum_{x=1}^{10} (x + c)^3$ where $c$ is a constant? [closed]

I'm kind of new to summations and wonder if there is a rule to sum cubic terms. For example $$\sum_{x=1}^{10} (x + c)^3$$ where $c$ is a constant. I know how to calculate this sum, but I wonder if ...
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1answer
21 views

Integer coefficients of cubic equation imply integer roots

Problem: Let $a,b,c$ be three integers for which the sum $ \frac{ab}{c}+ \frac{ac}{b}+ \frac{bc}{a}$ is integer. Prove that each of the three numbers $ \frac{ab}{c}, \quad \frac{ac}{b},\quad \frac{bc}...
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Challenger Theory Of Questions question

Let $y = P(x)$ be polynomial of degree $4$ such that $P(1) = 2$ and attains min value 1 at both $x=1$ and $x=2$. Then $P'(1)=$ a)$-1$ b)$2$ c)$0$ d)$1$ According to me graph of $4^{th}$ degree ...
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1answer
45 views

What is the integral of an inverse square root of a standard cubic formula?

$$\int\frac{1}{\sqrt{ax^3+bx^2+cx+d}}dx$$ I know the solution to the integral of an inversed square root of a quadratic equation. But I am not being able to solve this one and I have searched ...
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For which $a, b$ in $(-1,1)$ does $\frac{1}{3}x^3-a^2x+b=0$ have three real solutions?

I know $\frac{1}{3}x^3-a^2x+b=0$ will always have one solution as $\frac{-b}{x} = \frac{1}{3}x^2-a^2$ will always have one intersection in the upper two quadrants of the coordinate system. But I can'...
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1answer
70 views

Solving $x³+24x²+5x-13 = 0$ [closed]

I have been doing some mathematics and I have come across this question that I need to somehow solve to move onto the next questions. I'm currently undertaking an abstract algebra unit with this ...
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2answers
101 views

Solving $x^3 + 12x^2 + 8x − 7 = 0$

I have been doing some maths and I have come across this question that I need to somehow solve to move onto the next questions: $$x^3 + 12x^2 + 8x − 7 =0$$ Can anyone help and solve with a step by ...
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1answer
43 views

Cubic Discriminant Uses

The discriminant for cubic equations is - $Δ​\:=b^22\:^2−4ac^3−4b^3d−27a^2d^2+18abcd$ And I am aware that you can determine the number of roots a cubic has using method shown below - $Δ​\:>0$ ...
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2answers
62 views

Solve the Cubic $x^3+24x^2+6x-4$

I'm having trouble solving this cubic: $x^3+24x^2+6x-4$. Is anyone able to help explain how to get the values of $x$?
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109 views

Polynomial $x^3-2x^2-3x-4=0$

Let $\alpha,\beta,\gamma$ be three distinct roots of the polynomial $x^3-2x^2-3x-4=0$. Then find $$\frac{\alpha^6-\beta^6}{\alpha-\beta}+\frac{\beta^6-\gamma^6}{\beta-\gamma}+\frac{\gamma^6-\alpha^6}{\...
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51 views

Relation between the roots and coefficient.

Let Let a, b and c be the roots of the equation $$x^3 +3x^2-1=0$$Then what is the value of expression $a^2b+b^2c+c^2a$. I got it done by evaluate the sum and difference of $a^2b+b^2c+c^2a$ and $ab^...
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1answer
26 views

Polynomial how to factor $-3k^3-k^2+5 = 0$

After I got the determinant from a matrix this is what I was left with a cubic equation. How can I work with this? $-3k^3-k^2+5 = 0$ I have looked at various links and websites like this one https:...
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1answer
105 views

Does wolfram alpha gives all integers solution to $n=x^3+y^3+z^3$ if it has a finit solutions in integers? [closed]

I have tried to solve this equation in wolfram alpha $x^3+y^3+z^3=6112$ , I have got by wolfram alpha $x= 88,y=-34, z=-86$, But this equation have other solution which wolfram alpha didn't montioned ...
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31 views

Is there a generic way to solve cubic inequalities?

Hi Pardon me if it has already been asked. I could not find any material on math-stack, hence asked this question! I have a cubic inequality of the form, \begin{equation} -a_1x^3+a_2x^2+a_3x+a_4<...
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Does cubic equation have any integer solution for X if all constants are integers

I'm stuck with the following equation: $$x^3 + a \cdot x^2 + b \cdot x + c = d $$ $$ x, a, b, c, d > 1,$$ $$ x < d $$ $$ a < d $$ $$ b < d $$ $$ c < d $$ $$ x, a, b, c, d \in \Bbb{Z} ...
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124 views

Lagrange Method of Solving Cubic Equations

Let $K$ be a field (for simplicity, one may assume $K\subseteq\mathbb{C}$), and let $d\in K$. Denote $w=\sqrt[3]{d}$, and $\zeta=(-1+\sqrt{-3})/2$. If $f(x)=x^3+ax^2+bx+c$ is the minimal polynomial of ...
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73 views

A series of multiplication leads to $\frac{1}{2} = 2$

I'm presented with the equation $\frac{a+b}{a} = \frac{b}{a+b}$ Performing cross multiplication yields $a^2+2ab+b^2 = ab$ Subtracting $ab$ from both sides, we get $a^2+ab+b^2 = 0$ Multiplying both ...
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28 views

Necessary and sufficient condition for a cubic to have a root?

Let $k$ be a field, of characteristic different from $2$ or $3$. Let $p$ and $q$ be elements of $k$ and consider the equation $x^3 + px + q = 0$. If $q^2 + \frac{4}{27}p^3$ happens to be a square in $...
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58 views

Solving $\beta^3-2\beta^2+(1-\rho)=0$ for $\beta$

I need to solve the following equation for $\beta$: $$\beta^3-2\beta^2+(1-\rho)=0$$ where $\rho$ is just a constant. I already tried different kind of methods from the wikipedia page for cubic ...
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1answer
60 views

Is there anything special about this summation I found that calculates out the square, cube, fourth, etc. power of any integer?

let me start by saying that my formatting may be way off, but it's the best I can do, and has little to do with the question, and I will make sure I am as clear as humanly possible, including showing ...
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Finding roots using the cubic formula

I have obtained the cubic formula from google, for some generic cubic function: $$ax^3+bx^2+cx+d=0,$$ it is expressed: \begin{align*} x&=\sqrt[3]{\left(-\frac{b^3}{27a^3}+\frac{bc}{6a^2}-\frac{d}{...
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1answer
39 views

Exersise 7.3.(2) of Escofier Book

Let $a,b,c\in\mathbb{C}$ roots of $f(x)=x^3-3x+1$.Show that $\mathbb{Q}(a)$ is splitting field of $f(x)$ over $\mathbb{Q}$ and represent $b,c$ as linear combination of $\{1,a,a^2\}$.My question is ...
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1answer
23 views

Formula for 3 positive real roots of cubic, avoiding imaginary parts

I have a cubic equation, of which the roots are the eigenvalues of a 3x3 matrix. The coefficients depend on a parameter (z), and I want to prove that one of the eigenvalues ($\lambda_2$, the second ...
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1answer
68 views

Verify triginometric result of cubic equation $x^3-x^2-p^2x+p^2=0$

Consider the following cubic function, $$f(x):=(x+p)(x-p)(x-1)=x^3-x^2-p^2x+p^2$$ where $p\in(0,1)$ is a fixed parameter. Then the sum of the absolute value of the three roots is $$S_1:=1+2p$$ On the ...
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73 views

Given semiperimeter and radii of inscribed and circumscribed circles, find the side lengths of triangle

Consider $\triangle ABC$ with side lengths $a,b,c$, semiperimeter $\rho=\tfrac12\,(a+b+c)$, inradius $r$ and circumradius $R$. Let $u=\rho/R,\ v=r/R$, $a'=a/R,\ b'=b/R,\ c'=c/R$. Given $u,\,v$, we ...
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1answer
18 views

How can you deduce a cubic polynomial given two quadratic divisors, their respective remainders, and nothing else?

For example, A cubic polynomial gives the remainders $(5x + 4)$ and $(12x - 1)$ when divided by $(x^2 - x + 2)$ and $(x^2 + x - 1)$ respectively. How can I find the polynomial from this set of ...
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1answer
65 views

Show that $x^3+3y^3+9z^3-9xyz=1$ has infinitely many integer solutions. [duplicate]

Show that $x^3+3y^3+9z^3-9xyz=1$ has infinitely many integer solutions. I have found that (1,0,0) and (1,-18,12) are two solutions and tried (1,-18+n,12-n). There is a hint saying that I should try ...
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1answer
15 views

Multiple definitions of casus irreducibilis

In the case of cubic equations, Casus irreducibilis occurs when none of the roots is rational and when all three roots are distinct and real (...) —Wikipedia's Casus irreducibilis article So, $x^...
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2answers
71 views

Find the minimum value of $a^2+b^2+c^2+2abc$ when $a+b+c=3$ and $a,b,c\geq0$.

Given $a,b,c\geq0$ such that $a+b+c=3$, find the minimum value of $$P=a^2+b^2+c^2+2abc.$$ It seems like the minimum value of $P$ is $5$ when $a=b=c=1$, but I can find at least one example where $P<...
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Simple expression for root of specific cubic equation

In solving a cubic equation of the form $x^3-cx-\tfrac{4}{3}c$ with $c$ some function of a parameter $\mu$, I have encountered the following expression in one of the roots: $$\left(\frac{1+6\mu}{2+3\...
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2answers
63 views

If $\ x^3+px-q=0 $ has three roots $a$, $b$, $c$ then find an equation with roots $a+b$, $b+c$ and $c+a$

The roots are not $a$, $b$, $c$ but $\alpha$, $\beta$, and $\gamma$. I wrote $a$, $b$, $c$, due to space constraints. I know that I have to start with $$\ \alpha + \beta + \gamma = 0 $$ $$\ \alpha + \...
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2answers
61 views

Cubic with negative real roots — inequality on coefficients

Let $a, b, c \in \mathbb R$ such that all roots of $x^3 + a x^2 + b x + c$ are negative real numbers. If $a < 3$, prove that $b + c < 4$. My attempt: I tried first putting a as positive and ...
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0answers
41 views

Quadratic residue and cubic polynomials

The question is simple: Can $\sum_{x=0}^{p-1} (\frac{ax^3 +bx^2 +cx +d}{p} )$ be expressed as a closed form ? (It's a Legendre Symbol)
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110 views

$x^{3}+ax^2+bx+c$ has all roots negative real numbers and a<3. Establish an inequality between only b and c [duplicate]

A cubic equation $x^{3}+ax^2+bx+c$ has all negative real roots and $a, b, c\in R$ with $a<3.$ Prove that $b+c<4.$ My attempt : Let the cubic be $f(x)$ Plotting graph we see that , $f(x\...
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4answers
83 views

solution of the cubic equation $x^3 = 0$?

it is clear that one solution of the above equation is $x = 0$, but since it is a cubic equation so it should have $3$ complex roots, so what will be its other roots? Also how can we determine that a ...
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1answer
45 views

Solving the sextic equation with 14th root of unity

I am solving the sextic equation $t^6-t^5+t^4-t^3+t^2-t+1=0$ satisfied by the 14th root of unity (a problem from Ian Stewart's book). I was able to get up to the point where you have the polynomial $u^...
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1answer
53 views

Find $m^3$ if $m=\sqrt[3]{a + \frac{a+8}{3}\sqrt{\frac{a-1}{3}}} + \sqrt[3]{a - \frac{a+8}{3}\sqrt\frac{a-1}{3}}$

Please help me solve this question in a easy way: $$ \sqrt[3]{a + \frac{a+8}{3}\sqrt{\frac{a-1}{3}}} + \sqrt[3]{a - \frac{a+8}{3}\sqrt\frac{a-1}{3}} = m $$ Find $m^3$. (The answer is $8$.) I ...
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2answers
53 views

Calculating eigenvalues for a $3 \times 3$ matrix without solving a cubic

I am trying to find the eigenvalue for question (h), however I am unable to factor out and find the eigenvalues(roots) after I take the determinant of the characteristic equation. Let $x$ be an ...
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1answer
21 views

Simultaneous Equations Finding the Intersection of a Cubic and a Quadratic

My functions are y = -0.65(x -8.165)^2 + 1.5(x -8.165) + 6.872 y = 0.08(x-11)^3 -2.2(x-11) + 5.9 By using simultaneous equations and equating the functions to one another I've simplified it to the ...
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1answer
38 views

Estimating $f(x)=\frac{\sin(x)+\sec(x)+\tan(x)}{\cos(x)\csc(x)\cot(x)}$ where $x \in[-\frac{\pi}{3},\frac{\pi}{4}]$

Consider the function $$f(x)=\frac{\sin(x)+\sec(x)+\tan(x)}{\cos(x)\csc(x)\cot(x)}$$ in the interval $x \in[-\frac{\pi}{3},\frac{\pi}{4}]$. Find a combination of algebraic (not transcendental) ...
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2answers
77 views

Is Integer Solution of Cubic Negative?

I have been trying to solve a problem where OP seeks Integer roots to cubic equation. There are similar problems ( see 1, 2, 3, 4, 5, 6, 7) in this forum. To solve the general cubic$$x^3+ax^2+bx+c=0\...
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1answer
93 views

A proof of Poincaré-Hurwitz Theorem in English?

I was reading this paper by Shane Chern: https://arxiv.org/abs/1602.02844 and I found the following theorem: Let $E$ be a nonsingular cubic curve in $\mathbb{P}^2$ which is defined over $\mathbb{Q}$...
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5answers
288 views

Prove that if $2a^3 + 27c = 9ab,$ then the roots of $x^3 + ax^2 + bx + c = 0$ form an arithmetic sequence.

I am not sure how to begin this problem. Can someone help me? I have a hint: Let $y = x + \frac{a}{3}$ and rewrite $x^3 + ax^2 + bx + c = 0$ in terms of $y.$ How do I do this?
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2answers
86 views

Find all positive integer $a,b,c$ that $a^3+b^3+c^3$ can be divided by $a^2b,b^2c,c^2a$ [closed]

Find all triplets of positive integers $(a,b,c)$ for which $$a^3+b^3+c^3$$ is divisible by $a^2b$, $b^2c$ and $c^2a$. I just found that $a=b=c$ satisfies the problem. Are there any other ...
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1answer
66 views

Does every nonsingular cubic curve $C \subset \mathbb{P}^2$ over a finite field $\mathbb{F}_q$ have a flex over $\mathbb{F}_{q^3}$?

I cannot prove the following statement. Does every nonsingular cubic curve $C \subset \mathbb{P}^2$ over a finite field $\mathbb{F}_q$ have a flex over $\mathbb{F}_{q^3}$? Is there an elegant ...

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