All Questions
278
questions
2
votes
1
answer
56
views
Show that if $x=-1$ is a solution of $x^{3}-2bx^{2}-a^{2}x+b^{2}=0$, then $1-\sqrt{2}\le b\le1+\sqrt{2}$
$$x^{3}-2bx^{2}-a^{2}x+b^{2}=0$$
Show that if $x=-1$ is a solution, then $1-\sqrt{2}\le b\le1+\sqrt{2}$
I subbed in the solution $x=-1$, completed the square, and now I'm left with the equation $\...
- 602
0
votes
1
answer
43
views
Prove a relation between the coefficients of a depressed cubic.
The equation $x^{3}+px^{2}+q=0$ where p and q are non-zero constants, has three real roots $\alpha$, $\beta$ and $\gamma$. Given that the interval between $\alpha$ and $\beta$ is p and that the ...
- 21
1
vote
1
answer
45
views
Proof of conditions for polynomials
Find the conditions for the roots $\alpha, \beta, \gamma$ of the equation $x^3-ax^2+bx-c=0$ to be in: $(i)$A.P.; $(ii)$G.P.
If the roots are not in A.P. and if $\alpha+\lambda,\ \beta+\lambda,\ \...
- 87
0
votes
0
answers
125
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 ...
- 1
3
votes
6
answers
406
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 ...
- 167
3
votes
2
answers
222
views
Finding root of a cubic equation.
I was solving a physics statistical mechanics problem of an interacting system.
In that question, I have to find the eigenvalues of a matrix P whose elements are given by
$$P=
\begin{bmatrix}
e^{x} &...
- 217
6
votes
2
answers
176
views
Why do equilateral triangles relate to cubics
I found this question talking about the relation between an equilateral triangle and cubics with three distinct real roots.
Here's an image from the original post with an example:
What this post says ...
6
votes
5
answers
183
views
If the roots of $x^3 − 6x^2 + 10x + 1$ are denoted as a, b, c, then find the value of $(a^2 + b^2 )(a^2 + c^2 )(b^2 + c^2 )$.
If the roots of $x^3 − 6x^2 + 10x + 1$ are denoted as a, b, c, then find the value of $(a^2 + b^2 )(a^2 + c^2 )(b^2 + c^2 )$.
I have tried factoring $x^3 − 6x^2 + 10x + 1$ but didn't get anything. ...
5
votes
1
answer
127
views
Find the value of: $\sqrt[3]{a+b}+\sqrt[3]{b+c}+\sqrt[3]{a+c}$
Let $a,b,c$ be roots of the cubic
$$x^3-x^2-2x+1=0$$
Then, find the value of:
$$\sqrt[3]{a+b}+\sqrt[3]{b+c}+\sqrt[3]{a+c}$$
My attempt.
I used the substitutions $$a+b=x^3, b+c=y^3, a+c=z^3$$
$$x^3+y^...
- 652
0
votes
0
answers
105
views
When we say the odd degree polynomial has odd number of real roots, is there any condition on the coefficients?
I read that if the degree of a polynomial equation is odd then the number of real roots will also be odd.
I took the example of a cubic equation. If it has imaginary roots then that will occur in pair....
- 8,338
0
votes
3
answers
1k
views
Solve the equation $x^3-11x^2+38x-40=0$, given that the ratio of two of its roots is $2:1$.
Solve the equation $x^3-11x^2+38x-40=0$, given that the ratio of two of its roots is $2:1$.
By hit and try, I can see that $x=2$ is a root.
Dividing the given cubic by $x-2$, I get a quadratic whose ...
- 8,338
3
votes
3
answers
214
views
Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
Find the total number of roots of $(x^2+x+1)^2+2=(x^2+x+1)(x^2-2x-6)$, belonging to $(-2,4)$.
My Attempt:
On rearranging, I get, $(x^2+x+1)(3x+7)+2=0$
Or, $3x^3+10x^2+10x+9=0$
Derivative of the cubic ...
- 8,338
7
votes
3
answers
400
views
Reducing $ax^6-x^5+x^4+x^3-2x^2+1=0$ to a cubic equation using algebraic substitutions
Use algebraic substitutions and reduce the sextic equation to the cubic equation, where $a$ is a real number:
$$ax^6-x^5+x^4+x^3-2x^2+1=0$$
My attempts.
First, I tried to use the Rational root ...
- 652
3
votes
1
answer
107
views
Real solutions to the depressed cubic equation [closed]
How can I find the allowed domain to this depressed cubic inequality
$$x^3 - 3 x + 2 \cos(\frac{3 \sqrt{3} n}{2}) \geq 0$$
where $n$ is a real non-negative number. Using Cardano's method, I can obtain ...
- 403
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 ...
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