# Tagged Questions

Questions about the set of values at which a given function evaluates to zero. For questions about "square roots", "cube roots", and such, consider using the (radicals) and (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.

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### Express $c$ and $d$ in terms of $m$ where $c$ and $d$ are zeroes of $f$ where $m > -2$

Let $$f(x) = x^2 - mx -(6m^2+25m+25)$$ where $m > - 2$ It can be shown that $f(x)$ has two zeroes. Suppose we have $c,d \in \mathbb R$ s.t. $c < d$ and $f(c) = f(d) = 0$, express $c$ and $d$ ...
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### Finding the root of an indefinite polynomial

$0 = (a-n) x^{n-1} + ax^{n-2} + ax^{n-3} + \cdots + ax + a$ What is $x$ in terms of $a$ and $n$? I don't even know what this form of polynomial is called.
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### Are all the zeros of $1-a_2x^2+a_4x^4-a_6x^6+\cdots$ real for $a_{2n}>a_{2(n+1)}$ with $a_{2n+1}=0$ and $a_{2n}>0$?

This question is related to a previous question of mine. I was not pleased about the conditions I provided there. I had something different in mind but I failed in stating it. So here are the ...
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### How can I know how many real roots this equation has?

How many real roots does $2 \sin x-x=0$ have?
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### Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
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### Prove that $x^4+2x^2-6x+2=0$ when $x\in\mathbb{R}$ has exactly two solutions

Show that $x^4+2x^2-6x+2=0$ when $x\in\mathbb{R}$ has exactly two solutions. I first showed that the IVT guarantees that there exists at least one zero in $(0,1)$ and at least one zero in $(1,2)$. I ...
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### Why can a quartic polynomial never have three real and one complex root?

It seems that a quartic polynomial (degree $4$) either can have $0$ real, $1$ real, $2$ real, or $4$ real roots, and the rest is complex roots. Why can't it have $3$ real roots and $1$ complex?
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### How to identify properties of the zeroes of this polynomial? [on hold]

If $f_0(x)=1$, and $f_{n+1}=\frac{d}{dx}((x^2-1)f_n(x))$, prove that every $f_n$ has exactly $n$ distinct zeroes, all located in the interval $(-1,1)$. It's got me stumped, so any help/pointers would ...
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### Find $x_1^n+x_2^n$ on any quadratic equation, general case.

I have a simple quadratic (with $x^2$) equation, x can Be complex too: $$x^2+x+1=0$$ But it could be any equation, the equation above is just an example. I need to compute $x_1^{10}+x_2^{10}$, but ...
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### Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$

I've tried solving $\sqrt[3]{3} =a + b* \sqrt[3]{2}+c* \sqrt[3]{4}$, but there is no obvious contradiction, even when taking the norms/traces of both sides. I can't think of another approach. This is ...
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### Are there real solutions to $x^y = y^x = 3$ where $y \neq x$?

I need to solve the following equation for (x,y) $$x^y = y^x = 3$$ Everytime I run a numerical method for this problem, I get $$(x,y) = (1.82546...,1.82546..)$$ I expect there to be a solution ...
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### Surjectivity on the image of a annulus

I'm trying to prove the Fundamental Theorem of Algebra as it is done in Birkhoff and MacLane. Unfortunately, I don't have access to the book, only to a sketch. Therefore, I'm filling the gaps myself. ...
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### Find the relative width of a guitar fret

There is an equation to find the position of a fret on a guitar fretboard, given the length of a string is given by \begin{eqnarray} d = s – \frac{s}{2 ^ {(n / 12)}}, \end{eqnarray} where $d$ is the ...
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### Finding a value based on the roots of an equation

So I saw this question recently: Known $a^2+b^2+6a-12b+45=0$. Find $\dfrac{b-a}{b+a}$. I tried to factorize it but I don't really know how. Can someone help me with this?
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### Finding all real roots of an equation

I am looking for a computational method to find real roots of a function. The function looks like this: $$F(x) = \sum_{i=0}^n \frac{k_i}{\sqrt {(x - x_i)^2 + c_i^2}}.$$ I would like to use something ...
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### $F(x,t)=a_n(t)x^n+ \ldots +a_1(t)x+a_0(t)$. Show that $F(\cdot , t_0)$ has exactly one zero using the Implicit Function Theorem

$F(x,t)=a_n(t)x^n+ \ldots +a_1(t)x+a_0(t)$ is a through $t$ parametrized family of polynominals. $a_i : I \to \Bbb R \:\:\:\mathrm{ are }\: \mathcal C^k$- functions with $k \ge 1$. Let $x_0$ be a zero ...
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### Find m so that the equation has integer solutions

We are given the following equation: $(m+1)x^2-(2m+1)x-2m=0$, where $m\neq-1$. We have to find all integers $m$ so that the equation above has integer solutions. I know that $m=0$ and $m=-2$ satisfy ...
### Under what conditions will $x^2+bx+c=0$ have both roots real and positive?
Obviously, $x=\frac{-b\pm \sqrt {b^2-4c}}{2}$ and for real roots we must have $b^2-4c\geq 0$. But for what values of $a,b,c$ will the quadratic have both roots positive?