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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2
votes
1answer
42 views

What is the general solution of a multivariate quadratic equation

There exists a general solution to solve the general quadratic equation $$ax^2 + bx + c = 0$$ Solution: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ Does there exist a general solution for a general ...
8
votes
5answers
125 views

Proving that the roots of $1/(x + a_1) + 1/(x+a_2) + … + 1/(x+a_n) = 1/x$ are all real

Prove that the roots of the equation: $$\frac1{x + a_1} + \frac1{x+a_2} + \cdots + \frac1{x+a_n} = \frac1x$$ are all real, where $a_1, a_2, \ldots, a_n$ are all negative real numbers.
3
votes
6answers
120 views

Evaluate the expression $\sqrt{6-2\sqrt5} + \sqrt{6+2\sqrt5}$

$$\sqrt{6-2\sqrt5} + \sqrt{6+2\sqrt5}$$ Can anyone tell me the formula to this expression. I tried to solve in by adding the two expression together and get $\sqrt{12}$ but as I insert each ...
2
votes
3answers
58 views

How to prove that this quartic equation has exactly 2 real roots?

So I have this quartic equation here: $x^4-3x+1=0$ I'm supposed to prove this equation has exactly 2 roots. I defined $f(x)=x^4-3x+1=0$ Then I used the Intermediate value theorem at $f(0)$ and $f(...
3
votes
1answer
86 views

Finding the roots of an octic

I'm trying to solve a problem, but it involves finding the exact roots of the octic polynomial $$x^8+4x^7-10x^6-54x^5+9x^4+226x^3+125x^2-301x-269$$ How can I find the roots of an octic? Wolfram ...
1
vote
4answers
79 views

The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. [on hold]

The cubic equation $x^3-5x^2+6x-3 = 0$ has solutions $\alpha$, $\beta$ and $\gamma$. Find the value of $$\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}$$
1
vote
1answer
23 views

If $\alpha_i$ are the roots of $x^n + nax−b = 0$ then show that $\prod_{1< i \le n} (\alpha_1 -\alpha_i)=n(\alpha_1^{n-1}+a)$

If $\alpha_i$ are the roots of $x^n + nax−b = 0$ then I would like to show that $$\prod_{1< i \le n} (\alpha_1 -\alpha_i)=n(\alpha_1^{n-1}+a).$$ The only thing I could think is differentiating $x^...
-4
votes
0answers
20 views

Zeros of a polynomial. [on hold]

If $F(i, x) $ is a polynomial where $i$ is a parameter and $\rho$ is the largest root of $F(0,x)$ and $F(i+1,x)\ge F(i, x)$, Prove that as $i$ increases $\rho$ will increase. I don't understand ...
0
votes
2answers
19 views

Find $m$ so that given equation has real roots

We are given the following equation: $$x^4 - (2m - 1)x^2 + 4m - 5 = 0, m \in \mathbb{R}$$ Find all $m$'s so that the given equation has real roots. I thought I only had to put $\Delta \geq 0 $. ...
2
votes
3answers
69 views

Find the number of solutions of $ x^2+1=2^x$

I tried solving this equation $$x^2+1=2^x$$ but I was able to get only two roots , i.e. $x=0,1$ but the answer given said the equation has 3 roots when I looked at the graph given in the solution it ...
0
votes
1answer
35 views

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 $d$ ...
0
votes
2answers
27 views

Squareroot of complex number to the square $\sqrt{z^2}$

I have to calculate $\sqrt{z^2}$ an I am confused about how to procede. I thought about introducing $z=|z|\exp(i\phi+2\pi k) \implies z^2=|z|^2\exp(2i\phi+4\pi k)$. Hence, $$\sqrt{z^2}=\sqrt{|z|^2\...
0
votes
1answer
49 views
0
votes
3answers
67 views

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.
6
votes
2answers
280 views

Ways to find irrational roots of an n degree polynomial

I am trying to write a program to find the roots a given polynomial of degree N, with the form $$ A_{0}X^{N}+A_{1}X^{N-1}+A_{2}X^{N-2}+A_{3}X^{N-3}+...+A_{N} $$ I know that if there are rational ...
0
votes
2answers
42 views

Number of real roots of $\frac{a_1}{a_1-x}+\frac{a_2}{a_2-x}+…+\frac{a_n}{a_n-x}=2016$ for $0<a_1<…<a_n$?

Does it have exactly $n$ roots? Would replacing the R.H.S. of the equation with any other real number change the outcome? I can show that the equation has no complex roots. But how to find the number ...
1
vote
0answers
49 views

For what value of $(a+b)$ will all roots of $f(x)=x^4-8x^3+ax^2+bx+16$ be positive?

I was thinking of using Descartes' rule of signs, from which I find there are at most 2 positive roots and 2 negative roots of the given equation. Also, $f(\infty)>0$ and $f(0)>0$ imply that ...
0
votes
2answers
69 views

How can I know how many real roots this equation has?

How many real roots does $2 \sin x-x=0$ have?
1
vote
2answers
38 views

Find the stopping distance

(original question, see edits below for full context) After much frustration, I have figured out a function which maps velocity during acceleration/deceleration for my project. $$\text{velocity} =s+\...
1
vote
1answer
76 views

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 ...
-1
votes
0answers
32 views

Solve Equation with max integer [closed]

Solve please $\dfrac{\left[\sqrt{x-[x ]}\right]}{(x+3)(x+4)}\ \geq0$ edit
0
votes
1answer
44 views

Comparing the roots of two increasing functions

For any $0 \leq x \leq y \leq 1$, define $f(y;x):=\frac{y^2}{2}-\frac{2 y^3}{3}+\frac{y^4}{4} - \frac{x^2}{2} + \frac{x^3}{3}$ and $g(y;x):=\frac{y^2}{3}-\frac{2 y^3}{4}+\frac{y^4}{5} - \frac{x^2}{3} +...
2
votes
0answers
27 views

zeros of special sequence of polynomials

While answering this question, I make one question. Define a sequence of polynomials as \begin{align} p_{n}(x)=\sum_{r=0}^{\lfloor (n+1)/2\rfloor} (-1)^{r}\binom{n+1-r}{r} x^{n-r}. \end{align} I used ...
1
vote
3answers
73 views

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 ...
2
votes
0answers
20 views

How to identify properties of the zeroes of this polynomial? [closed]

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 ...
2
votes
3answers
83 views

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 ...
1
vote
0answers
35 views

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. ...
0
votes
1answer
20 views

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 ...
0
votes
1answer
37 views

Why has the equation positive root?

Let $x \in \mathbb{R}$ and $\lambda ,{\lambda _0} \in \mathbb{C}$ and $r\in(0,1)$. $w(x) = {\alpha _m}{x^m} + \cdots + {\alpha _1}{x^1} + {\alpha _0}$. $f(\lambda)$ is function such that $f(\...
3
votes
0answers
81 views

Is this equation $(n+1)~x^{2n+1}-n~x^{2n}-n=0$ solvable in radicals for some $n \geq 2$?

Consider this polynomial equation: $$(n+1)~x^{2n+1}-n~x^{2n}-n=0,~~~~n \geq 2,~~~n \in \mathbb{N}$$ It's related to another question of mine, but I don't think the context matters here. I'm ...
1
vote
2answers
66 views

Can a polynomial of $n$ degree have $n+1$ distinct real roots?

Question : Let $f(x) = \sum^n_{k=0}c_kx^k$ be a polynomial function then prove that if $f(x) = 0$ for $n+1$ distinct real values, then every coefficient $c_k$ in $f(x)$ is $0$ , thus $f(x) = 0$ for ...
0
votes
0answers
13 views

Connecting First Passage Time to Power Spectrum

Let $f$ be a real function. Is there a connection between The first positive abscissa for which its autocorrelation function is equal to zero (which I call the first passage time, fpt) The largest ...
-1
votes
1answer
30 views

If $w$ is an imaginary cube root of unity, then the polynomial whose roots are $2w+3w^2$ and $2w^2 + 3w$ is?

What polynomial with complex coefficients has the following as its roots? $2w+3w^2$ and $2w^2 + 3w$ I have tried doing this all the ways I know of, still can't get my pen over it... Can you guys ...
3
votes
0answers
35 views

how to find the the maximum of an implicit function

I have an implicit function and I would like to find the value of $h$ that maximizes $R$, i.e, I want to find $h$ that satisfies $\frac{\partial R}{\partial h} = 0$. The function is, $C=\frac{A}{1+\...
1
vote
3answers
64 views

Is there a difference between $\sqrt{x+2}+x=0$ and $x^2-x-2=0$

Is there a difference between $\sqrt{x+2}+x=0$ and $x^2-x-2=0$ Solutions are $x=2$ or $x=-1$. But $x=2$ does not satisfy $\sqrt{x+2}+x=0$.Because $\sqrt{4}+2 \neq0$ So does it mean that they are ...
0
votes
2answers
50 views

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?
2
votes
3answers
177 views

Hi! Just wondering if any one can help me out with this roots question? [closed]

(i). Factorise $z^2 - 5z + 6$ and hence, solve the equation $ z^2 - 5z + 6 = 0$ (ii). Show that $z^2 - 5z + 6$ is a factor of $z^3 + (-4 + i)z^2 + (1 - 5i)z + 6(1 + i)$. (iii). Find the three roots ...
6
votes
3answers
641 views

Finding the roots (contest math)

So the problem is : $x^4-4x^3-x^2-8x+4=0$, find all solutions A tip that I have gotten, is to divide both sides by $x^2$. I've tried so, but I do not manage to see any further. Do anyone know how ...
3
votes
0answers
28 views

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 ...
1
vote
1answer
35 views

$F(x,t)=a_n(t)x^n+ \ldots +a_1(t)x+a_0(t)$. 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 ...
0
votes
1answer
37 views

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 ...
1
vote
1answer
68 views

Simplifying $\frac{4\sqrt{7}}{3}\cos{\left(\frac{1}{3}\arccos{\frac{1}{\sqrt{28}}}\right)}+\frac{1}{3}$

I was finding the roots of the polynomial $y=x^3-x^2-9x+1$. And I got one of the roots of the polynomial to be $$\dfrac{4\sqrt{7}}{3}\cos{\left(\dfrac{1}{3}\arccos{\dfrac{1}{\sqrt{28}}}\right)}+\dfrac{...
0
votes
3answers
34 views

Localization of roots of complex quadratic equations

Let $a,b,c\in\mathbb C\setminus\{0\}$ be complex numbers such that $$b^2-4ac \neq 0.$$ We consider the equation $$ax^2+bx+c=0.$$ I am interested in general statements about the roots of this equation ...
0
votes
1answer
31 views

non-standard exponential-squared fog attenuation [closed]

I inherited a formula that I'm hoping to simplify. $d = \frac{\sqrt{-\log_2(t)}}{f\sqrt{\ln(2)}}$ Any ideas? Thanks, Jason EDIT (for context): This formula determines the exponent for exponential-...
0
votes
3answers
53 views

Have $n$ real root then show that ${(n-1)\left(c_{n-1}\right)^2} \geq 2 n c_{n-2} c_n$

What should I do here? I don't even know where to start from. Please help me by giving me a hint. If $$x^{n } - c_{1} x^{n-1}+c_2 x^{n-2} -c_3 x^{n-3}+\cdots+(-1)^{n-1} c_{n-1} x+(-1)^n c_n=0(c_1,...
4
votes
1answer
162 views

Find the roots of $e^x+e^{1/x} + a = 0$

Find the roots of this equation $e^x + e^{1/x} + a = 0$ where $a \in \Bbb R$ Is there any nice formula for this type of equation?
0
votes
4answers
50 views

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?
2
votes
1answer
22 views

Deciding if $i \in \mathbb{Q}(\alpha)$ for the root $\alpha$ of a certain polynomial

Consider the field $\mathbb{Q}(\alpha)$, where $\alpha$ is one of the (complex) roots of the polynomial $f(x) := x^3 + x + 1 \in \mathbb{Q}[x]$. I now want to find out if $i \in \mathbb{Q}(\alpha)$ ...
4
votes
2answers
47 views

Showing that $(x^2 - 2)(x^2 - 3)(x^2 - 6)$ has a root in $\mathbb{F}_p$

Let $p$ be a prime number, $K = \mathbb{F}_p$ the field with $p$ elements, and $f = (x^2 - 2)(x^2 - 3)(x^2 - 6) \in K[x]$. I now want to show that $f$ has a root in $K$. I know that to show the ...