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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.

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1answer
25 views

How to show the existence of a root for a specific equation?

There is an interesting question about the solvability of the following equation. Let $a, b, c, d$ be constant numbers. In addition, these constant real numbers satisfy exponent $a >1$, finite ...
1
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1answer
19 views

Solutions to a polynomial equation with constraint

I am looking for solutions to the following simple polynomial equation, $$ x_1^2 + x_2^2 + y_1^2 + y_2^2 = -2 (x_1 x_2 + 3 y_1 y_2) $$ where $x_i, y_i \in \mathbb{R}$. Importantly, I would like only ...
-6
votes
0answers
30 views

Find the roots of a quadratic [on hold]

1) Show that: $(a^2+1)(β^2+1)=(c−1)^2+b^2$ 2) Find In terms of b and c, a quadratic whose roots are $\frac a{a^2+1}$ and $\frac β{β^2+1}$ I do not know how to employ the first information in 1) to ...
0
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0answers
26 views

Order of poles on a function

How can I determine what the order of the pole on the following function is: $$\ f(z)= \frac{e^{bz}}{z\sinh(az)}$$ From the Laurent series, I found that the residue would be b/a or -b/a, however, I ...
1
vote
1answer
81 views

Prove that $e^{x}\frac{\text{d}^{n}}{\text{d}x^{n}}(x^{n}e^{-x})$ is a polynomial with positive zeroes. [on hold]

I need to prove that $$e^{x}\frac{\text{d}^{n}}{\text{d}x^{n}}(x^{n}e^{-x})$$ is a polynomial with positive zeros. I don't know how to approach to this problem. Need help
0
votes
3answers
57 views

solve $\cos(x)\cosh(x)-1=0$

I'm trying to find the limit value of this for large values of $x$, in terms of a closed form formula. However when I try to plot this using different representations I get different curves. For $\...
0
votes
1answer
32 views

How to find polynomial functions 3rd degree with no, one, two, three zeros(roots)?

I must find 3rd degree Polynomial functions in R[x] with: 1) no roots 2) only one root 3) only two roots 4) only 3 roots If the function has a root, then prove it. If not, then explain why. My ...
2
votes
2answers
29 views

Find the solutions of the equation

$x^3- 3x + 2 = 0$ Using the Horner scheme, I can find easily the roots of the equation: $x_1 = x_2 = 1 $ and $x_3 = 2.$ How can I find them using other method ?
1
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1answer
36 views

Solve an equation involving the error function

Let $0<a<1$ be given. The equation: $$a = 1 - \frac{2\sqrt{x/\pi}}{\mathrm e^x \mathrm{erf}(\sqrt x)}$$ has a unique root $x$, because the right-hand side is increasing in $x$, and goes to $0,...
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0answers
25 views

“Reverse” Vieta's Formulas

In one of my investigations, I needed to figure out sums of powers of roots of polynomial equations. These are not very hard to figure out. For monic polynomials, the first three sums are: ...
3
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1answer
59 views

Asymptotic behavior of roots of an equation involving exponential and logarithm

Prelude This Post is a continuation of this Original Post. The original problem asked is: How many solutions does the following equation have: $$ a^x = \log_a(x) \,,\quad a \in (0,1) \wedge x \in\...
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1answer
73 views
+50

A polynomial $p(x) \in \Bbb R_{2n-1}[x]$, $p(0) = 0$, $p(x) \geq 0 \ \forall x \geq 0$, can be written as $p(x) = xq_1(x)^2 + q_2(x)^2$

Let $p(x) \in \Bbb R_{2n-1}[x]$ be a polynomial such that $p(0)= 0$ and $p(x) \geq 0 \ \forall x \geq 0$. Then there exists $q_1, q_2 \in \Bbb R_{n-1}[x]$ such that $p(x) = xq_1(x)^2 + q_2(x)^2$. ...
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0answers
23 views

Finding Polynomial Roots Analytically

Let us consider the following function: $$f(s_1,s_2,...s_n,n)=\frac{1}{1+s_1}+\frac{1}{(1+s_2)^2}+...+\frac{1}{(1+s_n)^n},$$ where $s_i\in(0,1), i=1,...,n.$ Let us introduce the following function: $$...
0
votes
2answers
29 views

How to show that all the zeros of complex function are on unit circle.

Prove that if $be^{a+1}<1$ where $a$ and $b$ are positive and real, then the function $$z^ne^{-a}-be^z$$ has $n$ zeroes in the unit circle. I have no idea how to start this problem?
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4answers
1k views

Is it possible for the bisection method to provide “fake” zeros

I've read about the bisection method for finding roots of a function in my numerical analysis textbook and one question came to my mind. Given a relatively complicated function, the chances of ...
0
votes
0answers
58 views

Approximate a solution for $\frac{e^{-a}(v+u)(a^x+e^{a}x\Gamma(x,a))}{\Gamma(x+1)}-u\approx 0$

Is it possible to approximate (or even find) a solution for the following equation: $$\frac{e^{-a}(v+u)(a^x+e^{a}x\Gamma(x,a))}{\Gamma(x+1)}-u\approx0,$$ where $x\ge 0$ and integer, and the ...
1
vote
1answer
19 views

Finding the maximum of a function on a specific interval

I have a problem at it is as follows. I've to find the maxium value of the following function between to time points, namely $t=0$ and $t=\frac{2}{50}$. The function is the following: $$8-\exp(-\frac{...
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2answers
35 views

$p^3+q^3+r^3$, where $p$,$q$, and $r$ are the roots of the cubic function $x^3+4x^2-4x+1$. Working included.

I am trying to utilise the expressions for Vieta's fomulae to solve expressions, just as an investigation. The question I gave myself is, if $p$, $q$, and $r$ are the roots of a cubic, what is $p^3+q^...
2
votes
1answer
32 views

What's the intuitive explanation to how an equation has solutions in real or complex? Is there a “link” between real/complex?

I've been wondering, I've learned that equations may have solutions or roots in complex numbers. However, I've not entirely understood, how the transition between real and complex solutions occurs. ...
0
votes
1answer
40 views

How to graphically depict the possible solutions of a quadratic equation

I have the following quadratic equation : $$am^2 + bm + (c_1^2 +c_2^2) =0,$$ where the solution is given by $$m = \frac{-b\pm\sqrt{b^2-4a(c_1^2+c_2^2)}}{2a}.$$ Here, $\Delta>0$. Thus I have ...
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0answers
37 views

Roots of trigonometric equation $\sin((n+1)\theta) + 2\sin(n\theta) = 0,\;\;\theta\neq k\pi\,\,k\in\mathbb{Z}.$

I am trying to find a closed formula for the roots of the following trigonometric equation, $$\sin((n+1)\theta) + 2\sin(n\theta) = 0,\;\;\theta\neq k\pi\,\,k\in\mathbb{Z},$$ where $n$ is a positive ...
1
vote
2answers
57 views

Real solution of $(\cos x -\sin x)\cdot \bigg(2\tan x+\frac{1}{\cos x}\bigg)+2=0.$

Real solution of equation $$(\cos x -\sin x)\cdot \bigg(2\tan x+\frac{1}{\cos x}\bigg)+2=0.$$ Try: Using Half angle formula $\displaystyle \cos x=\frac{1-\tan^2x/2}{1+\tan^2 x/2}$ and $\...
1
vote
1answer
45 views

Approximate a solution for a single variable exponential equation

Can anyone please help me fined (if it is possible) a closed-form solution or an approximation for the solution for the following equation (x is the only variable): $$\frac{((a-1)b^{x+2}-(b-1)a^{x+2}+...
0
votes
2answers
47 views

How to make the nth root of a product act the same as simple multiplication in regard to parity?

I don't have any experience working with radicals, but I'm working on a function that requires products of nth roots to be positive or negative, depending on the number of negative factors. I've ...
1
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3answers
60 views

Finding the positive root of $x^3 +x^2 =0.1$ by numerical methods.

The positive root of $x^3 +x^2 =0.1$ is denoted to be $A$. $(a)$ Find the first approximation to $A$ by linear interpolation on the interval $(0,1)$ For this, I got $x_1 =0.05$ $(b)$ Indicate why ...
0
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1answer
38 views

Root of an quadratic equation

I have the following quadratic equation : $m^2 + m(p-1/l) - (\Omega_x^2 + \Omega_y^2)=0$ I would like to get the solution in terms of $\Omega_x, \Omega_y$ with some approximations i.e. neglecting $(...
-1
votes
1answer
21 views

Prime Factorization for Square Roots with unknowns

I need to help my daughter with math, but I don't understand it myself. We need to solve for $x$ and $y$ in the following equation, using prime factorization: $$\sqrt{1890x} = \sqrt{2100y}$$ Can ...
7
votes
2answers
388 views

Link between polynomial and derivative of polynomial

I can't seem to solve this problem, can anyone help me please? The problem is: Let real numbers $a$,$b$ and $c$, with $a ≤ b ≤ c$ be the 3 roots of the polynomial $p(x)=x^3 + qx^2 + rx + s$. Show ...
1
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1answer
56 views

If $f(x)$ has $n$ distinct roots in $R$, then $f'(x)$ has $n-1$ distinct roots in $R$ Without Rolle's Theorem

Prove that: If $f(x)$ has $n$ distinct roots in $R$, then $f'(x)$ has $n-1$ distinct roots in $R$ Without Rolle's Theorem. I know in this topic, There is proof with Rolle's theorem. It uses that if ...
0
votes
0answers
16 views

Number of zeros of $h(z)=3-z+2e^{-z}$ in the right half-plane $\Re(z)>0$ [duplicate]

I am supposed to show that $h(z)=3-z+2e^{-z}$ has exactly one zero in the right half-plane $\Re(z)>0$. I want to use Rouchés Theorem, which says that if $f$ and $g$ are analytic inside and on a ...
2
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2answers
52 views

Factoring cubic polynomial over R

$z^3-7z^2+14z-7=0$ I tried simplifying it to $z^3-7(z-1)^2=0$, but I don't think i can proceed from there. It sort of looked like geometric progression but it is not that either, and I don't see any ...
2
votes
3answers
77 views

Proof that the square root of a negative number is real.

So I stumbled upon this weird result when experimenting with fractional exponents. Suppose you have some negative, real number, for example -8. We know $\sqrt{-8}$ is not a real number.But $$\sqrt{-8} ...
1
vote
1answer
50 views

roots of cubic equation complex

Given the cubic equation: $$x^3-2kx^2-4kx+k^2=0.$$ If one root of the equation is less than $1$, another root is in the interval $(1,4)$ and the third root is greater than $4$, then the value ...
6
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5answers
116 views

Solve $\lim\limits_{n\to\infty}\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}$

I am having great problems in solving this: $$\lim\limits_{n\to\infty}\sqrt[3]{n+\sqrt{n}}-\sqrt[3]{n}$$ I am trying to solve this for hours, no solution in sight. I tried so many ways on my paper ...
0
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1answer
14 views

Irreducibility of a polynomial over Rationals with condition given on its coefficients.

Let $f = a_nX^n+\cdots+a_1X\pm p \in \mathbb{Z}[X]$ with $\sum_{i=1}^n |a_i| < p$. Show that $f$ is irreducible in $\mathbb{Q}[X]$. Hint: Show that every root of $f \in \mathbb{C}$ has modulus ...
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2answers
30 views

Factorise the following p(x) as the product of a linear term and a quadratic polynomial with no real roots.

Factorise the following p(x) functions as the product of a linear term and a quadratic polynomial with no real roots (or if there are real roots, factorise to irreducible form and find them): 1. $$p(...
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1answer
21 views

Combination of function's roots question

I have a trig function $$\sin(\frac{\pi}{3}x)$$ with roots at 0, 3, 6, 9. I also have a function $$\sin(\frac{\pi}{4}x)$$ with roots at 0, 4, 8, 12. I am looking for a generalized way to combine them ...
1
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1answer
43 views

“Fill in the Gaps” Trig function with integer zeros another trig function doesn't have

I have an interesting challenge involving roots of trig functions. I'm wondering if there is a method of creating a function that hits the integer roots that $$\sin(\frac{\pi}{5}x)*\sin(\frac{\pi}{3}x)...
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votes
2answers
52 views

Why isn't $i * i = 1$? [duplicate]

When we studied complex numbers they told us that $i * i = -1$ because $i = \sqrt -1$ and $i * i = i^2$, so the square removes the root. However we can say as well that $i * i = \sqrt {-1} * \sqrt {-...
0
votes
1answer
61 views

Solutions of $\cos(ax^c + bx) = 0$

As per title, I would like to find the zeros of $$ f(x) = \cos(ax^c + bx)$$ where $0\leq x \leq K$, $a \in \mathbb R$, $b \in \mathbb R$, and $c \in (0, 2]$. I have that $$ f(x) = 0 \Leftrightarrow ...
2
votes
2answers
27 views

Problem with the roots of polynomial given by sum of geometric series

Let's say I want to distribute leaflets for 60 months and in total I want to distribute $2500000$ leaflets. In my first month I want to distribute $8000$ leaflets and then I want to increase my speed ...
1
vote
1answer
37 views

Methods of determining if all roots of a polynomial have a negative real part

So my question is basically summarized in the title. The root of the question lies actually in application of the rules. Namely the stability of linear time invariant feedback systems is determined ...
1
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4answers
44 views

Find the range of values which has no real solutions

I would like to know how to solve the following problem: Find the range of values of the parameter $m$ for which the equation $2x^2 - mx + m = 0$ has no real solutions. I know I have to use the ...
0
votes
2answers
47 views

Does $5\sqrt{5}\div5\sqrt{5}$ equal 5 or 1 [closed]

Does $5\sqrt{5}\div5\sqrt{5}$ equal $5$ or $1$. I think it is $1$ but I just want to check I have not missed anything.
3
votes
5answers
111 views

General solution or approximate solution

Is there a known general or approximate explicit solution for $\xi$ in $$(1+\xi)^m (1-\xi)^n = C$$ where $m$ and $n$ positive fractions and $C$ being constant?
1
vote
2answers
80 views

Solving $8x^3 - 6x + 1$ using Cardano's method

Solve for the first root of $8x^3 - 6x + 1 = 0$ After solving I get $\sqrt[3]{\frac{-1 + \sqrt{3}i}{16}} - \sqrt[3]{\frac{1 + \sqrt{3}i}{16}}$, which is not a solution to the cubic equation: Here's ...
0
votes
1answer
26 views

Proof that if f(a)<0 and f(b)>0 f and is continuous on [a,b] then f changes sign at some c in (a,b) Part 2

I asked a question similar to this previously but I realized that what I was trying to prove was false. Then I changed the thing I was trying to prove but it was still false. I also accepted an answer ...
2
votes
3answers
235 views

How do we evaluate $(-i)^{\frac 5 2}$ without complex logarithms?

$$(-i)^{\frac 5 2} = (e^{-\frac{\pi i}{2}})^{\frac 5 2} = e^{-\frac{5\pi i}{4}}$$ How do we justify the last step before going to $e^{\frac{3\pi i}{4}}$? I think the approach is supposed to be $$(-i)...
-5
votes
1answer
37 views

Is $(-1)^{1/3} = -1$ , but $(-1)^{2/6} = 1$. Why aren't these the same? [duplicate]

So if you try to solve $(-1)^{1/3}$ you can do $(-1)^{1/3} = \sqrt[3]{-1} = -1$ (cubic root of $-1$) But what if I write $1/3$ as $2/6$? $$(-1)^{1/3} = (-1)^{2/6}$$ So $(-1)^{2/6} = \sqrt[6]{(-...
1
vote
2answers
38 views

Solution for $ y(x) - ae^{y(x)} = f(x) $ involving Lambert W function

I need to solve an equation of the type: $$y(x) -ae^{y(x)} = f(x)$$ with $a>0$. Furthermore, the expression for $f(x)$ can't be evaluated analytically (it's the solution of a differential ...