Skip to main content

Questions tagged [roots-of-cubics]

For questions related to roots of a cubic equation. All of the roots of the cubic equation can be found by the following means: algebraically, trigonometrically or numerical approximations of the roots.

Filter by
Sorted by
Tagged with
1 vote
1 answer
51 views

Cosine of a multiple of arctangent

I would like to find an expression for the following: $$\cos\left(\tfrac{1}{3} \arctan(B/A)\right)\\ \sin\left(\tfrac{1}{3} \arctan(B/A)\right)$$ This is a problem that comes up when trying to ...
Matthew Young's user avatar
-1 votes
0 answers
28 views

Solving a cubic with Cardano's method, unable to obtain the only irrational root. [closed]

Attempting to solve $5x^3 - 13x^2 + 18x - 20$ lead me to depress it to the $x^3 + px + q$ form, obtaining $5x^3 + \frac{101}{15}x - \frac{7364}{675}$. However, perhaps it might be the coefficient of $...
Spadester's user avatar
0 votes
1 answer
55 views

Relationship between roots and coefficients of a cubic

Let $f(x) =x^3-px+q,p>0,q>0$ and all the zeroes of $f(x) $ are real. Prove that if $\alpha$ be the root with least absolute value then $|\alpha|$ lies in the interval $(q/p, 3q/2p) $. I have ...
YBR's user avatar
  • 75
2 votes
1 answer
53 views

A question about Dummit & Foote's explanation on the resolvent cubic and the Galois group

I am at the beginning of my study of field theory and I am reading page $\textbf{615}$ of $\textbf{Dummit & Foote}$, and the part where I have question about is shown below: Here for part $\...
ZYX's user avatar
  • 1,131
1 vote
2 answers
62 views

Prove that $a=0$ if and only if $b=0$ for the cubic $x^3 + ax^2 + bx + c=0$ whose roots all have the same absolute value.

Take three real numbers $a, b$ and $c$ such that the roots of equation $x^3+ax^2+bx+c=0$ have the same absolute value. We need to show that $a=0$ if and only if $b=0$. I tried taking the roots as $p, ...
user1299519's user avatar
-3 votes
4 answers
124 views

Roots of $x^3-2=0$... [duplicate]

When I was solving this,$$\begin{equation}\begin{aligned} x^3-2&=0\\x^3&=2\\x&=2^{1/3}\end{aligned}\end{equation}$$,I got $x=2^{1/3}$ But this is only one root...I know there are two ...
math student's user avatar
  • 1,259
0 votes
0 answers
50 views

cubic equation edge cases

Working on general cubic equation solver in form ax^3+bx^2+cx+d=0 And have no clue for special cases: In terms of cubic there should be one real root and two complex, or 3 real roots if coefficients ...
Vitaly Protasov's user avatar
2 votes
3 answers
102 views

Prove that $a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$ has three real roots

I'm trying to prove that the cubic equation $a_3 \lambda^{3} + a_2 \lambda^{2} + a_1 \lambda + a_0 = 0$ has three real roots. The coefficients are $a_3 = - 1 - \sigma - \tau - \chi$ $a_2 = -2 (\sigma +...
Rich T's user avatar
  • 61
0 votes
1 answer
108 views

How to solve $x^3-3x+3=0$? [duplicate]

I have the following cubic equation. Is it possible to solve it analytically in very "easy" way to a student only has a pre-calculus level: The equation is: $$x^3-3x+3=0$$ The real root I ...
M.K.A.B's user avatar
  • 31
2 votes
1 answer
117 views

Three real roots of a cubic

Question: If the equation $z^3-mz^2+lz-k=0$ has three real roots, then necessary condition must be _______ $l=1$ $ l \neq 1$ $ m = 1$ $ m \neq 1$ I know there is a question here on stack about ...
Darshit Sharma's user avatar
0 votes
0 answers
38 views

inverting the first three elementary symmetric polynomials

Given the first three elementary symmetric polynomials $e_1 = x_1 + x_2 + x_3$, $e_2 = x_1 x_2 + x_1 x_3 + x_2 x_3$, $e_3 = x_1 x_2 x_3$ (where $x_1, x_2, x_3 \ge 0$), I wish to solve these equations ...
Tim02130's user avatar
0 votes
0 answers
42 views

Set of coefficients of degree three monic real polynomial with three real roots is connected.

Let $p(x)=x^3+ax^2+bx+c$ be a cubic polynomial with real coefficients $a, b, c,$ and define: $$D=\{(a,b,c)\in \mathbb{R}^3\mid \text{the polynomial}\ p(x)\ \text{factors into linear factors over }\ \...
nkh99's user avatar
  • 471
0 votes
2 answers
58 views

Finding roots of cubic equation by factorisation

I want to find solutions of a cubic equation which is in $x$. I know a cubic equation will have three roots. Consider this cubic equation $(a-xb)[(c-xd)(e-fx)-g]=0$ Here $a,b,c,d,e,f,g$ are constants. ...
Dinesh Katoch's user avatar
0 votes
1 answer
79 views

Summation of Roots of Cubic Equation

I was attempting the recent May/June 2023, CAIE past paper for Further Mathematics variant 12. The question states: 2     The cubic equation $x^3+4x^2+6x+1=0$ has roots $\alpha$, $\beta$, $\gamma$. ...
BeaconiteGuy's user avatar
1 vote
2 answers
140 views

How to solve $x^3−x+1=0$

I am interested in finding a solution for the equation: $$ x^3 - x + 1 = 0 $$ I've noticed that there are numerous polynomial equations where one of the coefficients is zero. Could you provide ...
winter's user avatar
  • 33
0 votes
0 answers
54 views

Primes $a$, $b$, $c$, $d$ satisfying $(\cos(2\pi/7))^{1/3}+(\cos(4\pi/7))^{1/3}+(\cos(6\pi/7))^{1/3}=\left(\frac{a-b\sqrt[3]c}d\right)^{1/3}$ [duplicate]

Source: Enumeration 2022 Prelims conducted by the Indian Institute of Science (question $10$ under objectives) Problem statement: Primes $a$, $b$, $c$, $d$ satisfy the following equation. $$\left(\...
Nothing special's user avatar
2 votes
1 answer
50 views

The set of coefficients of cubics having three real roots is connected

Let $D$ be the set of all $3-tuples$, $(a,b,c)$ in $R^3$ such that the cubic polynomial $x^3+ax^2+bx+c$ has three real roots, i.e., $D=\{(a,b,c) \in \mathbb{R}^3 \mid x^3+ax^2+bx+c \textit{ factors ...
nkh99's user avatar
  • 471
0 votes
0 answers
120 views

Can I use this algorithm for solving cubic equations?

I am trying to find the root solutions for a cubic equation including the eigenvalues of each root. I tried to put the equation into my calcualtor but the calculator doesn't show solutions that has ...
Ryan's user avatar
  • 1
1 vote
1 answer
106 views

Solving a cubic using triple angle for cos (i.e $\cos(3A)$)

a) Show that $x=2\sqrt{2}\cos(A)$ satisfies the cubic equation $x^3 - 6x = -2$ provided that $\cos(3A)$ = $\frac{-1}{2\sqrt{2}}$ I did not have a difficulty with this question, I have provided it for ...
Mikhael's user avatar
  • 15
1 vote
1 answer
200 views

How do I find a cubic equation given only one root?

Given the root of a cubic equation $Z = \sqrt[3]{Y + \sqrt{Y^2 - \frac{X^6}{27}}} + \sqrt[3]{Y - \sqrt{Y^2 - \frac{X^6}{27}}} - X$ and the assumption that both $X$ and $Y$ are greater than zero, is ...
Lawton's user avatar
  • 1,851
3 votes
3 answers
162 views

Is there any faster way to factor $x^3-3x+2$?

$$x^3-3x+2$$ $$x^3-3x+x^2+2-x^2$$ $$x^2-3x+2+x^3-x^2$$ $$(x-2)(x-1)+x^2(x-1)$$ $$(x-1)[x^2+x-2]$$ $$(x-1)(x+2)(x-1)$$ Is there a better, faster way to factor this cubic trinomial?
SirMrpirateroberts's user avatar
3 votes
6 answers
392 views

Find all real numbers $a$ for equation $x^3 + ax^2 + 51x + 2023=0$, has two equal roots.

Problem: Find all real numbers $a$ for which the equation, $x^3 + ax^2 + 51x + 2023=0$, has two equal roots. This problem is from an algebra round of a local high school math competition that has ...
JHumpdos's user avatar
  • 167
1 vote
0 answers
49 views

Order $3$ linear transforms invariating a binary cubic form

Consider $P(x,y)$ a homogenous polynomial of degree $3$ in two variables (a binary cubic). To it we associate first the $2\times 2$ matrix $$\frac{1}{2}\operatorname{Hess}(P) = \frac{1}{2}\cdot\left( ...
orangeskid's user avatar
  • 54.5k
3 votes
0 answers
64 views

Involution on monic cubic polynomials related to nesting/denesting of cubic radicals

Consider the involutive transformation $$\mathbb{R}^3 \ni (a,b,c) \overset{\phi}{\mapsto} \left( \frac{a + 2 c}{\sqrt{3}}, \frac{a^2 + a c + c^2}{3} - b , \frac{a - c}{\sqrt{3}}\right)$$ Show that if $...
orangeskid's user avatar
  • 54.5k
5 votes
0 answers
83 views

Involution on $2\times 2$ matrices

Show that the map on $2\times 2$ matrices \begin{eqnarray} \left( \begin{matrix} a & b\\ c & d \end{matrix} \right)\overset{\Phi}{\mapsto} \left( \begin{matrix} a & b\\ c & d \end{...
orangeskid's user avatar
  • 54.5k
2 votes
1 answer
109 views

Homographic relation between two roots of a cubic

Consider a cubic equation $ x^3 + 3a x^2 + 3 b x + c=0$ with distinct roots. Show that any two roots $x$, $y$ are connected by a homographic relation $$(a^2-b) x y + \frac{1}{2}\ (\ (a b-c+\delta) x +...
orangeskid's user avatar
  • 54.5k
1 vote
1 answer
67 views

Signs in the Cardano formula

When deriving the Cardano formula from $x^{3}+px+q=0$ we let $x$ be a sum and compare coefficients. So $x=u+v$, then we get a system for $u$ and $v$. We get $(1) -q=u^{3}+v^{3}$ and $(2) u^{3}v^{3}=-(\...
thereisnoname's user avatar
3 votes
2 answers
114 views

$\sqrt[3]{x_1} + \sqrt[3]{x_2} + \sqrt[3]{x_3} = \sqrt[3]{z}$ if $x_i$ are the real distinct roots of $(x+y)^3 - x^2 z + f x z( x + y + f^2/27 z)$

Show that $$\sqrt[3]{x_1}+ \sqrt[3]{x_2} + \sqrt[3]{x_3} = \sqrt[3]{z}$$ where $x_1$, $x_2$, $x_3$ are the real distinct roots of a cubic polynomial in $x$ of the form $$(x+y)^3 - x^2 z + f x z \left(...
orangeskid's user avatar
  • 54.5k
2 votes
2 answers
201 views

$\sqrt[3]{x_1} + \sqrt[3]{x_2} + \sqrt[3]{x_3} = \sqrt[3]{a},\,$ if $x_i$ are the real roots of $(x+b)^3 - a x^2$

Consider the equation, $$(x+b)^3 = a x^2\tag1$$ with $a\ne 0$, and real roots $(x_1$, $x_2$, $x_3)$. Show that, \begin{eqnarray}\sqrt[3]{x_1} &+& \ \sqrt[3]{x_2} &+ &\ \sqrt[3]{x_3} &...
orangeskid's user avatar
  • 54.5k
4 votes
5 answers
191 views

Solving a tricky equation $4x^3-5x^2-5 = 0$

How does one solve $4x^3-5x^2-5 = 0$? I've tried the substitution $y = x - \frac{5}{12}$ but then I ended up with this monster of an equation: $864y^3 - 450y -1205 = 0$. Now I'm stuck. Any help would ...
jkfthd hiifj's user avatar
4 votes
1 answer
314 views

Solution to depressed cubics

First of all I wanted to clarify that this is my first post here. I was trying to find a solution to the general depressed cubic polynomial and was able to get to the right formula but there are some ...
asd's user avatar
  • 43
-1 votes
1 answer
86 views

Solving a cubic equation in exact terms which is the key to solving the question in picture below [closed]

I have arrived at the formula below, I need help with solving this cubic equation for h please.
Nimna De Silva's user avatar
0 votes
0 answers
56 views

Hudde's cubic proof

I've been following the proof of Hudde's description of Cardano's method of cubic roots shown here https://proofwiki.org/wiki/Cardano's_Formula. Does anyone know where this proof comes from? I can't ...
Bountifull's user avatar
1 vote
0 answers
141 views

Roots of a polynomial equation.

I have been solving the cubic polynomial equation. The simplest of this is $x^3-1=0$ On solving this equation considering only the real values, I got only one solution i.e $x=1$ But a cubic polynomial ...
Jaya Guru's user avatar
0 votes
0 answers
85 views

geometric solution to cubic equations

Consider the cubic equation $x^3+d=bx^2$ with $ b,d > 0 $. The question is to give a geometric solution to this equation by interesting two conic sections. In class, our teacher showed us how to ...
user19170731's user avatar
4 votes
3 answers
137 views

Algebraically Solve $\left[a + b\sqrt{57}~\right]^3 = 540 + 84\sqrt{57}.$

Unclear how valuable this posting is. It really should be limited to specifying that the goal is to denest one level of the radicals, in an expression like $$\left[c + d\sqrt{D}\right]^{1/3} + \left[...
user2661923's user avatar
  • 36.9k
4 votes
2 answers
200 views

How can we use Cardano's method to solve a real life problem?

I am making a math project for my school. We can make it on any topic, but should involve some college level math. I have chosen 'Cardano's method' as my topic. I will be showing the method to solve a ...
Deepani Agarwal's user avatar
1 vote
2 answers
81 views

How to determine the order of the real roots of a cubic equation?

This is a self-answered question (I didn't find a reference, and thought of documenting this). Consider the equation $$ t^3+pt+q=0. $$ Its discriminant is $$ \Delta=-(4p^3+27q^2). $$ Suppose that it ...
Asaf Shachar's user avatar
  • 25.3k
1 vote
3 answers
91 views

How many roots of $x(1-x)^{2}=s$ are there in $(0,1)$?

This is a self-answered question, which is part of answering this related question. Alternative solutions are welcomed. Let $0<s < \frac{4}{27}$. Prove that the cubic equation $x(1-x)^{2}=s$ has ...
Asaf Shachar's user avatar
  • 25.3k
1 vote
2 answers
86 views

Need help understanding cubic formula derivation by Daniel Rui

I am reading a cubic formula derivation here: http://danielrui.com/papers/cubicPolynomial.pdf It looks fairly straight forward. The author defined: $y = \sqrt[3]{u} − \sqrt[3]v$ so far so good, I ...
some user's user avatar
  • 111
0 votes
1 answer
122 views

Further information on the reduction of cubic equations to a system of two conic sections

This question follows on from one I have previously asked, How to separate cubic equations into two conic sections: Deep dive into Omar Khayyam and I now would like some further advice on some ...
Bountifull's user avatar
5 votes
2 answers
360 views

How to separate cubic equations into two conic sections: Deep dive into Omar Khayyam

Lots of people have asked how to use Khayyam's method but I am studying for my dissertation so really need to understand the why. What I really don't understand/ can't find useful proofs for is how he ...
Bountifull's user avatar
4 votes
1 answer
250 views

Is there the continuous real root of the cubic equation and is there a closed formula to present it?

For any cubic equation, $ax^{3}+bx^{2}+cx+d=0$, we know there is always a real root if $a,b,c,d$ are all real. Suppose that $a,b,c,d$ are continuous and real function with respect of $i\in \mathbb{R}$,...
Lee White's user avatar
1 vote
1 answer
58 views

Range of an equation $y=\frac{(x-\alpha)(x^3-3x+1)}{x-\alpha}$

If the range of $y=\frac{(x-\alpha)(x^3-3x+1)}{x-\alpha}$ is all real numbers, then number of integers in the range of $\alpha$ is (1) 2 (2) 3 (3) 5 (4)Infinite I have no idea of how to do these type ...
Samar Imam Zaidi's user avatar
2 votes
0 answers
79 views

Deforming roots of a cubic polynomial

This is problem 9-1 from Milnor, dynamics of one complex variable (arxiv). Let $f_{\alpha}(z) = z + \alpha z^2 + z^3$. Show that $f_{\alpha}$ can be perturbed so that the double fixed point at the ...
dummy's user avatar
  • 571
3 votes
3 answers
147 views

Let $r,s,t$ are roots of the cubic equation $x^3+bx^2+cx+d=f(x)$ then write down $D=((r-s)(r-t)(s-t))^2$ in terms of $b, c, d$

Let $r,s,t$ are roots of the cubic equation $f(x) = x^3+bx^2+cx+d$ then write down $D=((r-s)(r-t)(s-t))^2$ in terms of $b,c,d$. Is there any clever way to solve it? We know $r+s+t=-b, rs+st+rt=c, rst=...
Ri-Li's user avatar
  • 9,088
2 votes
3 answers
175 views

If $f(x) =ax^3+bx^2+cx+d$ is a cubic equation with roots $\alpha,\beta,\gamma.$ Is there a way to find $\alpha^2\beta+\beta^2\gamma+\gamma^2\alpha?$ [duplicate]

Suppose $f(x) = ax^3 + bx^2 + cx + d$ is a cubic equation with roots $\alpha, \beta, \gamma.$ Then we have: $\alpha + \beta + \gamma= -\frac{b}{a}\quad (1)$ $\alpha\beta + \beta\gamma + \gamma\alpha = ...
Adam Rubinson's user avatar
0 votes
3 answers
96 views

Finding all possible roots of the equation

Find all possible solutions to the equation $$(x^3-x)+(y^3-y)=z^3-z$$ where $(x,y,z)\gt1$ and $\in\mathbb{Z}$ and not all three of them are equal. The original question didn't have the last condition ...
abcdefu's user avatar
  • 850
0 votes
0 answers
73 views

What would the cubic formula be if roots never existed? [duplicate]

$$ax^{2}+bx+c=0$$ $$x=-\frac{b}{a}+\frac{1}{\frac{b}{c}+\frac{1}{-\frac{b}{a}+\frac{1}{\frac{b}{c}+\frac{1}{...}}}}$$ $$ax^{3}+bx^{2}+cx+d=0$$ $$x=?$$ I know it can't recur like the quadratic ...
Baby Hearty Bear's user avatar
0 votes
3 answers
75 views

How to solve polynomials

Solve the equation $64x^3-240x^2+284x-105=0$ given that the roots are in an arithmetic . I tried having the roots as $a, (a+d), (a+2d)$ Factorising out $a$, $a(1+d+2d)$
Pascal Mathew's user avatar