Questions tagged [algebraic-equations]

Use this tag for questions related to solving equations involving polynomials.

0
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3answers
87 views

Algebraic equation of first degree with absolute values [closed]

How to solve $$\left| a^2-2 a-b^2-4 b-x \right| + \left| a^2-2 a-b^2-4 b-3 x+2\right| +\\ \left| a^2-2 a+b^2+4 b+2 x \right| + a^2-2 a+b^2+4 b+18 \left| x-2 \right| +11 x=20 $$ over the reals if ...
0
votes
1answer
29 views

Solve the equation for x.

solve the following equation for x $(x^2-8x-3)÷(8x-3)=(x^2+4x+4)÷(4x+4)$ The problem here I am encountering with is figuring out which method to be used here for solving. I am not getting how to ...
0
votes
3answers
50 views

Solve $(1+x)^2=A\sqrt{1+Cx}$ for $x$.

$x>0$, $A>0$ and $C>1$. I am trying to come up with a closed form expression for $x$, even if it is an approximation. Any help appreciated.
2
votes
1answer
49 views

how to handle a “Stiff” algebraic equation numerically?

I have a question of great practical importance for me, but I would like to ask it on a bit more of a theoretical mode, because I feel I lack the basic knowledge on it. I would also like to mention ...
3
votes
1answer
119 views

Solving $8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$

Solve the equation $$8x-3+\sqrt{x+2}-\sqrt{x-1}=7 \sqrt{x^2+x-2}$$ I have this idea: set $$\sqrt{x+2}=a , x+2=a^2 , \sqrt{x-1}=b.$$ So $$x-1=b^2 , 2a^2+6b^2 =8b-4$$ and $$x^2+x-2 =a^2b^2$$ and ...
1
vote
0answers
42 views

If $x$ is algebraic over a quotient field $K$ of $A$, then there exists an integral element $cx$ for some $A \ni c \neq 0$.

Let $A$ be a commutative ring, $K$ its quotient field and $x$ algebraic over $K$. This means that there exists a polynomial $f(X)$ with coefficients in $K$ such that $f(x) = 0$. In other words, if ...
0
votes
1answer
26 views

upper and lower limits for finding just one (real) solution of an algebraic equation (degree 5 and lower)

While programming an equation solver (using trial and error), I came across the fact that there are multiple real and complex bounds in which all of the solutions should be. For my case, only real ...
2
votes
1answer
48 views

Solution to $Mx-x^2=0$ where $x^2$ is the square of the elements of vector $x$

I have been trying to find the solutions for $$Mx=x\circ x$$ where $\circ$ is the element wise product. One solution is $x=0$. But there is another solution $x\neq 0$, if $M$ has a real positive ...
0
votes
3answers
75 views

Solving algebraic equations involving factorials - without trial and error.

The question is defined as follows: $$\frac{(x!)^3}{x}-1=3455$$ I first did the basics which was getting rid of the 1 onto the left and getting rid of the $x$, then I factorised both sides to get an ...
4
votes
1answer
125 views

Applying the quadratic Tschirnhausen transformation

As per my previous question, I attempted to take dxiv's approach, though I can't seem to make much headway. Considering the simpler problem $x^3=x+a$ and the substitution $y=x^2+mx+n$, I got the ...
1
vote
1answer
58 views

Proving non-existence of real roots

Prove that for any real numbers $a_{85}, a_{84}, a_{83},\dots ,a_3$ the equation $a_{85}x^{85}+ a_{84}x^{84}+ a_{83}x^{83}+\dots +a_3x^3+3x^2+2x+1=0$ has no real roots. (This problem is stated in ...
0
votes
4answers
66 views

Solving a symmetric equation involving three variables a,b and c

Solve : $$\frac{x+a^2}{a+b}+\frac{x+b^2}{b+c}+\frac{x+c^2}{c+a}=2(a+b+c)$$ I am trying to find a simple technique to solve this equation as there is a pattern in the equation, but I could not do. Any ...
0
votes
1answer
52 views

approximating irrational roots of algebraic equations with the Pierce expansion

Let $ p (x) = 1 - x \lfloor \frac{1}{x} \rfloor $ then the Pierce expansion of a real number $x \in {R}$ is expressed by \begin{equation} x_1 = \sum_{n = 1}^{\infty} (- 1)^{n + 1} \prod_{m = 1}^n ...
0
votes
2answers
114 views

Solving $x^2 y''(x) + 6x y'(x) - 9y(x) =0$ with similar techniques that are used to solve algebraic equations

Consider the ODE $$x^2 y''(x) + 6x y'(x) - 9y(x) =0 .$$ It is clear that we can solve the ODE by the method of reduction of order. However, if we "see" the function $y$ as some constant just for a ...
1
vote
0answers
20 views

Does this equation imply separation of variables?

I have the following equation: \begin{equation} \dfrac{q(t,r)}{s(t,r)}=B(r), \qquad \qquad (1) \end{equation} with $q>0$ and $s>0$. Apart from the trivial case $q(t,r)=\mathrm{const.} \times ...
0
votes
1answer
12 views

Equations of Value Unknown interest rate

i have two question in some assessments, both based on "Equations of Value Unknown interest rate" and two irregular contributions. I've been over the textbook, and looking along time on line found ...
0
votes
3answers
22 views

Is $-1$ sum of squares in the field $\mathbb{Q(\beta)}$ where $\beta = 2^{1/3}e^{2\pi i /3}$

Prove that $−1$ is not a sum of squares in the field $\mathbb{Q(\beta)}$ where $\beta = 2^{1/3}e^{2\pi i /3}$ My attempt : In fact $\Bbb Q(\beta)$ and $\Bbb Q(2^{1/3})$ are naturally isomorphic. So I ...
-3
votes
2answers
40 views

Two problems on solving equations in rational and integers [closed]

prove that $x^2-2y^2=7$ has infinitely many solutions in integers. $a$ and $b$ are two non-zero rational numbers.Then if the equation $ax^2+by^2=0$ has a nonzero rational solution $(x_0,y_0)...
4
votes
1answer
341 views

Explanation of the Tschirnhausen transformation

I am studying the resolution of the quintic equations, which involves the so-called Tschirnhausen transform. The idea is to cancel the fourth and third degree coefficients by a change of variable of ...
3
votes
2answers
196 views

How to find all nonnegative integers $x, y, z$ and $w$ such that $2^x3^y-5^z7^w=1$

Find all nonnegative integers $x, y, z$ and $w$ such that $2^x3^y-5^z7^w=1.$ I think they are $(x,y,z,w)=(1,0,0,0),(1,1,1,0),(3,0,0,1),(2,2,1,1)$, but I couldn't prove its sufficiency (or there may ...
1
vote
1answer
132 views

Find positive integer solutions of cubic equation with three variables

Find positive integer solutions of equation $$t^3 -at^2 + bt - c=0,$$ where $$a^3-6ab+7c=0$$ ($a, b, c$ are positive integers too). I've tried to use find solutions in modular arithmetic, but there ...
1
vote
2answers
52 views

Show that $\frac {1}{\sqrt{5}}[(\frac {1}{x+r_+}) - (\frac {1}{x+r_-}) = \frac {1}{\sqrt{5}x}[(\frac {1}{1-r_{+}x}) - (\frac {1}{1-r_{-}x})] $

I need to manipulate this equation: $$ \frac {1}{\sqrt{5}}\left(\frac {1}{x+r_+} - \frac {1}{x+r_-}\right) $$ to show that $$ \frac {1}{\sqrt{5}}\left(\frac {1}{x+r_+} - \frac {1}{x+r_-}\right) = \...
0
votes
1answer
57 views

Is it constant?

Let $G$ be an abelian group, and let $n:G\times G\rightarrow \mathbb{Z}$ be a map satisfying for all $u,v,w\in G$ $$ n(u+v,w)+n(u,v)=n(u,v+w)+ n(v,w)$$ and $n(0,x)=n(y,0)=0$ for all $x,y\in G$ Is $n$...
0
votes
1answer
63 views

Simplify algebraic expression with radicals: $\frac{1- ax}{1+ ax} \cdot \sqrt\frac{1+bx}{1-bx}$

I got stuck trying to simplify this roots forest: $\frac{1- ax}{1+ ax}*\sqrt\frac{1+bx}{1-bx}$ where x= $\sqrt{\frac{2a}{b}-1}$ So it is: $\frac{1- a\sqrt{\frac{2a}{b}-1}}{1+ a\sqrt{\frac{2a}{...
0
votes
2answers
76 views

A simple fraction with a complex answer: possibly right under my nose

I've been doing some calculations in fractions, and found this equation pop up to calculate my answer: $$\frac{1-x}{1+x}=x$$ the initial equation is $$\frac{2(x-1)}{\frac{4(x+1)}{2}}+x=4x+9(-4x-2)-...