Questions tagged [roots]
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 the (arithmetic) tag. For questions about roots of Lie algebras, use the (lie-algebra) tag instead.
6,782 questions
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Newton-Raphson Method for one equation and two variables
It is easy to visualize that $(2,3)$ is the root of the function $f(x_1,x_2)=(x_1-2)^2+(x_2-3)^2$.
I want to solve $f(x_1,x_2)=(x_1-2)^2+(x_2-3)^2=0$ using Newton-Raphson method starting from, say, $\...
- 1
-1
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2
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70
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Newton Method counterexample Polynomial
Is there an example of a polynomial that converges very slowly with Newton algorithm and very fast with jenkins-traub algorithm?
- 306
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3
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104
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Find $m \in \mathbb{Z} $ so that all roots of $x^2-mx+m+2=0$ are also integers.
I need a little help on this. I know $\Delta=m^2-4m+4, \Delta\ \ge0$ because I need my roots to be integers. From this one i get $$m\in(-\infty,2-2\sqrt{3}] \cup [2+2\sqrt{3},\infty).$$ Also $S=m, P=m+...
- 301
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1
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62
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Real-life example of fail by not using equivalent transformation [closed]
Solve the equation :
$\sqrt{x}=x-2$
Square it :
$x=(x-2)^2$
Expand :
$x=x^2 - 4x +4$
Everything to the left side :
$x - (x^2 - 4x +4) = 0$
Refine the equation :
$x^2 - 5x +4 = 0$
Factorize :
$(x-1)(x-...
- 335
0
votes
1
answer
78
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Poles of general function $\frac{1}{1+z^n}$
I am investigating the integral $\int_0^\infty \frac{1}{1+x^n}dx$ for $n>1$ and am currently looking at the case when $n$ is rational. To do this, I am considering the slice contour with angle $2\...
- 147
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54
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Relation between roots of finite sum of exponentials and the finite sum of their their geometric series on the positive real line
In my research, I am dealing with functions
$$f(x) = \sum_{k=1}^N \frac{A_k}{B_k} \left(1- e^{-B_k x} \right), \quad g(x) = \sum^N_{k=1} \frac{A_k}{B_k}\left( \frac{1}{2}- \frac{1}{1 + e^{-B_k x}} \...
- 23
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0
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55
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What is the principal n-th root of a complex/real number?
When taking the square root of a real number, we take the principal (positive) value. For example, if we have:
$$
\sqrt{x} = y \in (0, \infty) \implies y^2 = x
$$
Additionally, we know that:
$$
\sqrt{...
- 11
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votes
1
answer
80
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Please explain the additional PEMDAS rules for rooting a negative?
In addition to the traditional rules of PEMDAS, there appear to be additional rules we must know when rooting a negative. For instance, a well known example is:
$$
\sqrt{(-1)(-1)} = \sqrt{1} = 1
$$
...
- 7
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1
answer
58
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Polynomials of arbitrary degree with at most 3 real roots
Playing around with polynomials of the form $ax^n(1-x)+b(1-x)-2$ it seems that they have at most 3 real roots, for any value of $(a,b)$ and any natural $n$.
How to prove this?
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2
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How to find total real roots of $x^4+4x^3+12x^2+7x-3$
Question
The number of real values of x that satisfies the equation:
$$x^4+4x^3+12x^2+7x-3=0$$
Let $f(x)=x^4+4x^3+12x^2+7x-3$
My Approach
$f'(x)=4x^3+12x^2+24x+7$
$f''(x)=12x^2+24x+24$
$f''(x)=12(x^2+...
- 505
2
votes
1
answer
60
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Solving a recurrence relation using roots of a quadratic equation
if the roots of the quadratic equation $x^2 = px + q$ are $\alpha,\beta$. since they are roots,
$$\alpha^2=p \alpha+q$$
$$\alpha^n=p \alpha^{n-1}+q \alpha^{n-2}$$
similarly can be proven from $\beta$, ...
- 121
2
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0
answers
91
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Meromorphic function $g$ sharing value(s) with its derivative $g'$ counting multiplicity (CM). What does it mean?
I am working on some of the fundamental results of functions sharing values with their derivatives as an application of Nevanlinna Theory. There are many results for meromorphic functions sharing ...
- 63
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55
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Estimating the root of the equation $𝑎^𝑥+𝑏^𝑥=𝑐^x$ for triangle with sides $𝑎,𝑏,𝑐$
Let $(𝑎,𝑏,𝑐)$ be the sides of a triangle of circumradius $𝑅$. By the roots of a triangle we mean the positive root of the exponential equation $𝑎^𝑥+𝑏^𝑥=𝑐^x$. Clearly for the root to exist, we ...
- 22.1k
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2
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94
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Complex Roots for Parabolic-- Alternative Method
Lol I am kind of embarrassed how long I was thinking about this-- any help or added insights would be awesome!!!
Let's say we have a problem like:
If it says determine the complex Roots from the graph
...
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42
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Lower-bound on a root of a polynomial $px^a - x^b + (1 - p)$
I have a polynomial $px^a - x^b + (1 - p)$, where $a > b \in \mathbb{N}$ and $\frac{b}{a} < p < 1$ to make this polynomial has two roots on (0; 1]: $x = 1$ and another root. I want to find a ...
- 21
2
votes
1
answer
75
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Particular Polynomial Vanishing Mod p
Suppose that $q$ is a prime power, and let $\mathbb{F}$ be the field with $q^2$ elements.
Suppose that
$i,j \in \mathbb{Z} / (q^2 - 1)\mathbb{Z}$
$(q-1) \mid (i-j),$
$(q+1) \nmid i,j$.
Then I am ...
- 1,700
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1
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66
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Method to solve $0= \frac{x-1}{x} + f(x)$
Given an equation of the form:
$$0= \frac{x-1}{x} + f(x)$$
Knowing what $x_0$ makes $f(x_0)=0$, is it possible to determine when the full equation vanish? Are there particular techniques to study such ...
- 83
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1
answer
48
views
Linearizing around a point in system of nonlinear differential equations
Consider nonlinear system for $X(t)$ and $Y(t)$:
$$X'(t) = X(t) + Y(t) - 2$$
$$Y'(t) = 3 - X(t) Y(t)$$
subject to initial condition $X(t) = Y(t) = 0$. Computer package (like Mathematica) gives the ...
- 659
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votes
1
answer
39
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Newton-Raphson Principle
In the following picture, there are typo in notations. The Y-axis will be $t(x)$, and on the X-axis, the first point is $x^m$ and the 2nd point is $x^{(m-1)}.$
My question is I didn't understand the ...
- 115
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1
answer
103
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Let $a,b,c,d$ be real roots of $x^4-px^3+qx^2-px+1=0$ $(a>b>c>d>0)$. Prove that $ad=bc=1$.
Let $a,b,c,d$ be real roots of $x^4-px^3+qx^2-px+1 = 0\; (a>b>c>d>0).$ Prove that $ad=bc=1$.
My approach: $abcd=1$, and
$$
a+b+c+d = \frac1a + \frac1b + \frac1c + \frac1d,
$$
which can ...
- 295
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1
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Relationship of average root of functions to root of average function
Let $f(x, a) = ( a x - 1 ) / ( \exp( a x ) - 1 )$. Note that $f$ has a root at $x = 1 / a$.
Let $x^\star$ be the root of $f(\cdot, a_1 ) + f( \cdot, a_2)$, where $a_1, a_2 > 0$. By plotting and ...
- 115
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1
answer
63
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Every Quadratic in $\mathbb F_{5}$ has a root [closed]
I saw somewhere that every quadratic in $\mathbb F_{5}$ has a root. How do you prove this?
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84
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Why is one of the roots of this quadratic so well approximated by power mean?
I have a quadratic of the form:
$$y=x^{2}(1+n)-xn(a+b)-ab(1-n)$$
and I realized that the root that I'm interested in is equal to or well approximated by the power mean of $a$ and $b$
$$x=\left(\frac{a^...
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0
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31
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Does there exist analytic expression for the root of Bessel functions?
The root of the transcendental equation $x =\cos x$ has analytic expression: $$
D=\frac{1}{\pi} \int_0^\pi \arctan \left(
\tan \left(
\frac{t-\sin t +\frac{\pi}{2}}{2}
\right)
\right) dt + \frac{1 }{...
- 51
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1
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Help understanding the difference between solving inequalities when squaring them
Just wanted help understanding what is the fundamental difference between these following 2 ways of solving the inequality:
$$
2-\sqrt{x-1} < 1
$$
When I went on to solve it like:
$-\sqrt{x-1} < ...
2
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1
answer
167
views
Are there functions analogous to $\zeta(s)$ where some non-trivial zeros are known to be off the critical line?
Are there examples of functions $f(s)$ with functional equation $$f(s)=g(s)\, f(1-s)$$ analogous to the Riemann zeta function $\zeta(s)$ with an infinite number of non-trivial zeros which are limited ...
- 8,141
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1
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Roots of polynomials with reciprocal coefficients
Let $\mathbb{C}\ni a_0,\cdots,a_n\ne 0$ and define the polynomials
$$
p(x) = \sum_{k=0}^n a_k x^k,\,p^{*}(x)= \sum_{k=0}^n a_k^{-1} x^k
$$
I'm interested in when the roots of $p$ and $p^{*}$ coincide; ...
- 8,214
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1
answer
45
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Continuity of a root of a sequence of polynomials when extending one of the exponents of the polynomial sequence to be real
Consider the polynomial
$$
x^n+a x+b
$$
Suppose that I showed that for some regime of $a$ and $b$, the polynomials $x^n+a x+b$ have a unique positive root (the polynomial sequence provided is just an ...
- 307
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vote
1
answer
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Determining The Domain of a Function With a Fractional Exponent
This is quite a rookie question, I realize, and I feel a little bit stupid for asking this, but I'm really confused.
Basically, is this $$\left(\sqrt[3]{x - 3}\right)^{2}$$
equal to this $$\sqrt[3]{(...
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1
vote
1
answer
59
views
A question about complex algebraic numbers as roots of polynomials.
Suppose I have the root of a polynomial that has integer coefficients, i.e., it is a complex algebraic number. Can it ever be of the form $x = \alpha + \tfrac{p}{q}i$ where $\alpha$ is the ...
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Roots of trigonometric equation [duplicate]
I’ve been trying to find a way to solve this function for the first positive root. Solving this analytically seems to be above my high school education. Any and all help is greatly appreciated :)
The ...
- 1
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2
answers
92
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Solving an exponential and logarithmic equation with Lambert W function
I would like to solve this equation
$$\left(\frac{1}{50}\right)^x=\log_{\frac{1}{50}}x$$
I plot graphs on GeoGebra and I found that there are three intersections (i.e. solutions) from this equation. ...
- 11
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votes
2
answers
150
views
What is the new best known bound on the number of Riemann zeta function zeros with real part 3/4?
According to the popular news articles at https://www.science.org/content/article/sensational-breakthrough-marks-step-toward-revealing-hidden-structure-prime-numbers and https://www.quantamagazine.org/...
- 6,388
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1
answer
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Is these inequalities a criteria that a real polynomial has only real roots?
Suppose $f(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_0\in\mathbb R[x]$ is a real polynomial with indeterminates $a_0,\cdots,a_{n-1}\in\mathbb R$.
Let $z_1,\cdots,z_n$ be all complex roots of $f(x)=0$.
Consider ...
- 868
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0
answers
56
views
Irreducibility for polynomial $-x^p+ax+b$ if $a|b$
Motivated by the research I did to find an answer to my question here about integer roots of some polynomial, I discovered the Eisenstein criterion on the coefficients of a polynomial.
Consider a ...
- 1,355
1
vote
1
answer
36
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Tournament of Towns Problem about determining coefficients of a Polynomial
Baron Munchhausen was told that some polynomial $P(x) = a_nx^n + . . . + a_1x + a_0$ is such that $P(x) + P(−x)$ has exactly 45 distinct real
roots. Baron doesn’t know the value of $n$. Nevertheless ...
- 337
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0
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54
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Zeroes of $\Im(\text{cd}(x|i))+\Re(\text{cd}(x|i))$
Can someone help to determine the zeroes of this function in $0 < x < 10$ range? I need the analytical expressions, not numerical values.
$$\Im(\text{cd}(x|i))+\Re(\text{cd}(x|i))$$
here $\text{...
2
votes
3
answers
199
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Why is square root symbol the principal square root?
When researching square roots I found that $\sqrt{x}$ is the principal square root and $\pm\sqrt{x}$ is the square roots, with the reason for why being given through an example equation by user9464 of:...
- 31
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Forming a cubic polynomial with sum of pair of roots as roots. Broken.
Today I found something weird,
I don't think there is a careless mistake in my steps, yet I also cant believe this loop hole.
Btw. note this is not a homework question, the question I am asking came ...
- 52
1
vote
0
answers
56
views
Complex integration and root-finding algorithms
I try to evaluate an integral of the form
\begin{equation}
I(R) = \int_{-R}^{R}\frac{1}{\left(f(x)+e^{i\phi(x)}\right)\left(f(x)+e^{-i\phi(x)}\right)}\,\mathrm{d}x
\end{equation}
where $f\in\mathbb{R}...
- 313
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votes
1
answer
103
views
Find fixed point iterations $x=g(x)$ to find roots of $0=f(x) = e^x + x^2 - 4$
I am finding trouble looking for 2 possible $g(x)$s to use for the fixed point iteration of $$f(x) = e^x + x^2 - 4$$
So far, I have guessed that the solution lies in the interval $[-2, 1]$, and I have ...
- 57
1
vote
0
answers
79
views
A quartic formula over a finite field.
This is a follow up question to this:
In general, how does one solve a quartic equation over a finite field?
The Question:
What is the formula for $x$ in $\Bbb F_q$ when $$f(x)=ax^4+bx^3+cx^2+dx+e=0,$...
- 47.2k
0
votes
0
answers
95
views
Prove Steffensen's method converges quadratically
Steffensen's method is defined as $x_{n+1} = x_n - \frac{f(x_n)}{g(x_n)}, g(x) = \frac{f(x+f(x)) - f(x)}{f(x)}$
Suppose the sequence converges to $x^{*}$, then to prove quadratic convergence it must ...
- 327
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votes
1
answer
94
views
Representation of a function defined on the unit disk
Show that if $f(z)$ is continuous for $|z| \leq 1$ and analytic for $|z|<1$, and if $|f(z)|=1$ for $|z|=1$, then $f(z)$ can be written as
$$ f(z) = e^{i \varphi} \displaystyle \left(\frac{z-a_1}{1-\...
- 385
5
votes
1
answer
169
views
Roots of polynomials defined by the Syracuse (Collatz) sequence
I'm a french graduate student, and I stumbled on a problem which seems to surpass my current abilities...
My goal was to study polynomials defined by the Syracuse sequence (or Collatz sequence). By ...
2
votes
0
answers
67
views
how many roots can we have for equation $e^{a_1x}+e^{a_2x}+...+e^{a_nx}=ne^{cx}$?
Solve equation: $$\sum_{n\in\{1,...m\}}(-1)^ne^{-c_nx}=0$$ where $c_n$ are positive constants.
Is it true that such equation can have at most two real roots?
Sum of exponential functions have at most ...
- 862
1
vote
2
answers
110
views
Using Newtons method to find multiple polynomial roots
How is Newtons method used to find multiple polynomial roots?
I’m reading here on Wikipedia that “When one root r has been found, one may use Euclidean division for removing the factor x – r from the ...
- 143
9
votes
1
answer
663
views
In general, how does one solve a quartic equation over a finite field?
My Galois theory is a little rusty, so this might be a quick question for you to solve. There doesn't seem to be anything about this on MSE (based on the search "[galois-theory] [finite-fields] ...
- 47.2k
5
votes
2
answers
87
views
Quadratic with integer coefficients and $m$, $n$ such that $f(m)=n$ and $f(n)=m$
Let $a,b,c\in\mathbb{Z}$ and the function $f:\mathbb{R}\to\mathbb{R}$, $f(x)=ax^2+bx+c$. Given that there exist two distinct integers $m$ and $n$ such that $f(m)=n$ and $f(n)=m$, prove that the ...
- 1,053
2
votes
1
answer
95
views
The zeros of $f(s,a) = \sum_{n=1}^{a} (\frac{n^2 + n}{2})^{-s} $
I was looking at the zeros of
$$
f(s,a) = \sum_{n=1}^{a} \Big(\frac{n^2 + n}{2}\Big)^{-s}
$$
for integer $a>3$ in the strip $0 < \operatorname{Re}(s) < \frac{1}{2}$, and of course of the ...
- 17.1k