Questions tagged [algebraic-equations]

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

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1
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1answer
32 views

Need help simplifying a set of equations (and understanding how to solve it)

i have three algebraic expressions, each using the others. in these equations a, b, c and <...
3
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4answers
253 views

Roots of a certain sixth order polynomial

I am looking for the roots (or basically any information regarding them) of the sixth order polynomial $$p(x):=ax^6+(a+1)x^4+2bx^3-b^2$$ for positive, real constants $a,b$. Since $p(0)=-b^2<0$ and $...
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0answers
33 views

Reduction of a system of multivariate nonlinear polynomials

I am trying to find a way to simplify a set of multivariate polynomial equations. As an example, given the free variables $\mu_1,\mu_\star,x,\psi_1,\psi_2$, I am specifically interested in the ...
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3answers
80 views

How to solve Quadratic Equations with an Unknown C and other variables.

For instance $3x^2 - 11x + r$, I understand the value of $r$ is $6$ through trial and error but trial and error is extremely inefficient and time consuming thus not useful in exam situations, How ...
10
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2answers
370 views

What's wrong with manipulating this algebraic equation? and why does a manipulated system of equations have a different solution than the original?

I'll give an example for my first question: $x^2 + x + 1 = 0$ Clearly $x = 0$ and $x = 1$ aren't solutions, so first we can safely divide by $x$: $x + 1 + 1/x = 0$ By subtracting $1/x$ from both sides ...
4
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2answers
124 views

Solve $2x^2+y^2-z=2\sqrt{4x+8y-z}-19$

I am trying to solve the following equation. $$ 2x^2+y^2-z=2\sqrt{4x+8y-z}-19 $$ To get rid of the square root, I tried squaring both sides which lead to $$ (2x^2+y^2-z+19)^2=16x+32y-4z $$ which was ...
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1answer
40 views

Simplify $a.b+c.d$

Suppose that, $(S, +, \cdot)$ is a semiring, where the operations are defined as $x\cdot y=min(x, y)$ and $x + y=max(x, y)$. Can we further simply the expression $a\cdot b+c\cdot d$?, where $a, b, c, ...
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1answer
20 views

Resolve 1 equation with 3 unknowns with specific condition for each unknown

I want to find the possible solution of the following problem: $T_k= 21600$ seconds $1281.49 < T < 21600 $(T in seconds) $x$ and $y$ don't have unit $x$ and $y$ have to be whole number $x > 0$...
2
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1answer
51 views

If $\sum_{i=k}^n {n \choose i} p^{i}(1-p)^{n-i} \approx 0.05$, how can we find $k$?

Let $n$ be any natural number, let $k\in\{0, \dots, n\}$, and let $p \in [0, 1]$. If $\sum_{i=k}^n {n \choose i} p^{i}(1-p)^{n-i} \approx 0.05$, how can we find $k$ (in terms of $n$ and $p$)?
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0answers
28 views

Expanding algebraic equations of polynomials.

So suppose we have $G(z+h,w+t)$ where $G$ is an algebraic equation such that $G(z,w)=g_0(z)+g_1(z)w+\cdots+g_m(z)w^m$, each $g_i(z)$ is a polynomial of $z \in \mathbb{C}$, and $w$ is any function. My ...
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1answer
41 views

Solve $3^x+x^3=17.$ [duplicate]

It's a question I find a bit difficult to solve, and the question is: $$3^x+x^3=17.$$
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0answers
31 views

How can a field L be a subgroup of the Galois group of the extension L:K?

I am studying the notion of a Galois group, and a remark in my notes is the following: "$L:K$ an extension. Let $H$ be a subgroup of $Aut(L)$. Let $L^{H}$ be the fixed field of $H$. If $L \leq G(L:...
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1answer
106 views

Show that, if a, b, c are real numbers and ac = 2(b + d), then at least one of $x^2 + ax + b = 0$ and $x^2 + cx + d = 0$ has real roots.

Show that, if $a, b, c$ are real numbers and $ac = 2(b + d)$, then, at least one of the equations $x^2 + ax + b = 0$ and $x^2 + cx + d = 0$ has real roots. I've have tried many times and used ...
0
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1answer
71 views

How to find the roots to the polynomial $z^3+z^2+iz^2-z+1+3i$

I'm having a bit of a problem on how to compute forward the values for z from this equation: $$ z^3+z^2+iz^2-z+1+3i=0 $$ In the question there's given that one solution or root is z=i, by solving ...
0
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1answer
38 views

vertical displacement

I have the below HW question: What I've done is replace t with 2, thus obtaining that the ball is at a height of 1.25m after 2 seconds and hence, it will travel over the net. Am I correct?
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0answers
38 views

On multiple roots in algebraic closure of a given field

I’m having trouble digesting a fact that popped up in a proof from Lang’s Algebra, page 247, proposition 6.1. Given an element $\alpha$ algebraic over $k$, and its minimum polynomial $f(X)$ over k, ...
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1answer
17 views

Problem Following Algebra Steps

I am trying to follow along an answer to a particular question and don't understand one of the steps taken. I don't understand how they got from step 3 to step 4? The steps can be seen below. I ...
1
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2answers
30 views

Problem Following Along Algebra Steps

I have been given a question with the solution. However I am having trouble following the maths steps that have been taken in the solution. I'm unsure how they have gotten from step 1 to step 2, ...
3
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4answers
102 views

Is there a better way to solve this equation?

I came across this equation: $x + \dfrac{3x}{\sqrt{x^2 - 9}} = \dfrac{35}{4}$ Wolfram Alpha found 2 roots: $x=5$ and $x=\dfrac{15}{4}$, which "coincidentally" add up to $\dfrac{35}{4}$. So I'm ...
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2answers
104 views

Why the definition of Galois group in rotman’s Advanced modern algebra not considering repeated roots

The definition is provided in the upper figure. What if f(x) had repeated roots? Are the roots considered deduplicated such that there’ are less than n roots f(x) have? As @reuns pointed out, I have ...
0
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2answers
51 views

Algebraic solution for a system of algebraic equations?

I was asked about solving algebraically the following system of algebraic equations. $$f(a,b):=a(1-b)+ab\frac a{a+b}.$$ $$u = f(a,b),\quad v = f(b,a).$$ Solve algebraically $(a,b)$ in terms of $(u,v)$ ...
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1answer
51 views

The real numbers $a,b,c,d$ satisfy $a^2+b^2+c^2+1=d+\sqrt{a+b+c-d}$. Find the value of $d$. [duplicate]

The real numbers $a,b,c,d$ satisfy $a^2+b^2+c^2+1=d+\sqrt{a+b+c-d}$. Find the value of $d$. I've been given this question for a class I'm taking and I'm not really sure where to start. I let $\sqrt{a+...
0
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1answer
37 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
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3answers
54 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
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1answer
63 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
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1answer
140 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 ...
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0answers
47 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
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1answer
71 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
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1answer
51 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
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3answers
80 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
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1answer
294 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
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1answer
64 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
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4answers
143 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
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1answer
67 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
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2answers
132 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
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0answers
21 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
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1answer
21 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
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3answers
47 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 ...
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2answers
49 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
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1answer
1k 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
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2answers
235 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
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1answer
219 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
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2answers
55 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) = \...
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1answer
60 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
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1answer
68 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
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2answers
84 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)-...