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Tagged with cubics systems-of-equations
73
questions
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Solving a mixed system of 2 cubic and 2 quadratic equations with 4 unknowns
I tried plugging these cubic and quadratic equations into Wolfram Alpha and Symbolab but both said the same thing, too much computing time required. Now I am struggling to solve these equations and I ...
- 1
4
votes
1
answer
259
views
Why doesn't simultaneous equations work to find co-efficients of a cubic that passes through four points?
I'm trying to find the equation of a cubic that passes through three specific points (technically it's four but that point is y-intercept). The equation would look something like this:$f(x)=ax^3+bx^2+...
- 43
1
vote
1
answer
479
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Solving Cubic Systems of Diophantine Equations
What techniques are there for solving systems of Cubic Diophantine equations? I know there is no general purpose technique and looking at some papers it can quickly go over my head even for just a ...
- 13
2
votes
2
answers
152
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Solving the system $x^3+y= 3x+4$, $2y^3+z = 6y+6$, $3z^3+x=9z+8$
Solve the system $$\begin{equation} \label{equation1}
\begin{split}
x^3+y= 3x+4 \\
2y^3+z = 6y+6 \\
3z^3+x=9z+8
\end{split}
\end{equation}$$
By the theorem of triviality, I assumed $x=y=z=k$ and ...
- 860
1
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0
answers
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Solving a Type of System of Cubic Equations
Is there a closed-form solution to the following type of system of cubic equations?
$$
x_j=\sum_{i=1}^na_{ij}x_i^3,\quad\forall\,j=1,...,n
$$
Here the $a_{ij}$'s are constants, and the $x_i$'s are the ...
- 709
2
votes
3
answers
158
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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 ...
- 21
4
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2
answers
116
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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?
- 281
1
vote
2
answers
85
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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 ...
- 11
-1
votes
1
answer
150
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Solving Non-Homogenous Recurrence Relation
I was interested in Solution of this Non-Homogenous Recurrence Relation
$f(n)=f(n-1) + f(n-3) + 1$
The Base conditions are:
$f(0)=1$
$f(1)=2$
$f(2)=3$
Kindly help me in solving this Recurrence ...
- 1,143
5
votes
1
answer
244
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Cubic polynomials with distinct integer roots
Consider the polynomials $x^3+ax^2+c$ and $x^3+cx^2+a$ where $a,c\in\mathbb Z$. Is it possible that both of the polynomials have three distinct integer roots? If yes, find such $a$ and $c$ such that $\...
- 3,966
8
votes
3
answers
385
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Solving a scary looking cubic by hand
I came across this question in the Resonance Journal of science education (April edition 2021). Unfortunately it is only available in the hard copy of the magazine. There is a small poem cited from $...
- 426
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votes
1
answer
54
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Confusion in a multivariable cubic polynomial.
WARNING:-
This question may have wrong tags.
HISTORY:-
After watching an old numberphile video(https://youtu.be/wymmCdLdPvM)(this video has nothing to do with my question) ,I got interested in the ...
- 2,469
1
vote
1
answer
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System of Equations $ 2x ^ 3 + x + 2y - y ^ 3 = 0$, $4x ^ 3 - 2xy + 3y ^ 2 = 10$
I have found a system of equations which I cannot solve.
$$\begin {cases}
2x ^ 3 + x + 2y - y ^ 3 = 0 \\
4 x ^ 2 - 2xy + 3y ^ 2 = 10 \\
\end {cases}
$$
I noticed that the second equation can be ...
- 941
1
vote
2
answers
53
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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'...
- 317
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2
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92
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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^...
- 53