Roots Definition and 962 Threads

I was wondering if scientists or mathematicians have any use for complex numbers involving negative roots of I as in i=(-1)^(1/2). but my question is more what would be (-1)^(-1/2)?

I think I'm a bit rusty here, started with finding poles for $ \frac{1}{{z^4}+4} $, {z: |z-1| LE 2} 1) Out of interest, is there a complex equivalent of the rational roots test? The above function is obvious, but for a poly that has both real and complex roots? 2) I am using the exponential...

$a=1,2,3,4,5,------2011$, the roots of the equations $x^2-2x-a^2-a=0,$ are : $(\alpha_1,\beta_1),(\alpha_2,\beta_2),----------,(\alpha_{2011},\beta_{2011})$ respectively please find : $\sum_{n=1}^{2011}(\dfrac{1}{\alpha_n}+\dfrac {1}{\beta_n})$

The full question is: "How can we take square root of both sides of an inequality or equation just by multiplying each side by numbers with negative rational exponents". I will include several examples to explain how I think about it. 1)a=b, a^(-0.5)*a=b*a^(-0.5) (but a^(-0.5)=b^(-0.5)) then...

Looking for someone to check my working & answers please. Problem is 'find all the zeros of sin z' A) sin z = sin(x+iy) = sin(x)cosh(y) + i cos(x)sinh(y) Roots are when sin(x)cosh(y) = 0 = cos(x)sinh(y) $If \: sinh(y)=0, then \: cosh(y)=1 \: (cosh^2 - sinh^2=1) $ $ \therefore sin(x) = 0...

Homework Statement Express (-1)1/10 in exponential form (My first time posting - I hope I got the syntax right!) Homework Equations The Attempt at a Solution [/B] I got the solution, it's ejπ/10, but I'm not sure why. Here's my work: (-1)1/10 = (cos(π) + jsin(π))1/10 = cos(pi/10) +...

Homework Statement Solve the initial value problem Homework Equations Quadratic Formula The Attempt at a Solution My problem is that I don't understand how to solve the constants now, I understand, 2 equations, 2 unknowns, but when I plug the y(0) = 0 into the YsubH equation...

Is any Irrational Roots Theorem been developed for polynomial functions in the same way as Rational Roots Theorems for polynomial functions? We can choose several possible RATIONAL roots to test when we have polynomial functions; but if there are suspected IRRATIONAL roots, can they be found...

Sum the series below: $$\sum_{n=1}^{999}\dfrac{1}{a_n}$$ where $a_n=\sqrt[3]{n^2-2n+1}+\sqrt[3]{n^2+2n+1}+\sqrt[3]{n^2-1}$.

I've been studying for my final exam, and came across this homework problem (that has already been solved, and graded.): "Show that the Galois group of ##f(x)=x^3-1## over ℚ, is cyclic of order 2." I had a question related to this problem, but not about this problem exactly. What follows is...

Homework Statement Without solving the equation 3x^2-8x-3=0 show it has 2 different rational roots.[/B]Homework Equations ax^2+bx+c=0 The Attempt at a Solution I would appreciate if someone would check my work, and advise if I have done the right or wrong thing? Thank you, Jaco [/B]...

Let F be a field extension of Q (the rationals) with [F:Q] = 24. Prove that the polynomial $$x^5+2x^4-16x^3+6x-10$$ has no roots in F. Proof: Let $$a$$ be a root of $$x^5+2x^4-16x^3+6x-10$$. Since the polynomial has degree 5 by theorem we know that $$[Q(a):Q]=5$$. If $$a \in F$$ and...

Dear Friends! I need to find roots of polynomials with variable coefficients, The command I used is w=0:50 A=w^2 B=w^3+2 C=w+2*w^2 D=w E=w./2 ss=[A B C D E] xi=roots(ss) by this I find all the roots of equation, I want to find velocities by setting v1=w/xi(1) v2=w/xi(2) v3=w/xi(3) v4=w/xi(4)...

I'm homeschooled, but it's gotten to the point that my Mom doesn't know how to do what she's teaching me anymore. So now I'm teaching myself with just a textbook and no one to explain it to me. I'm stuck on an issue probably simple, but I still need help. I believe I messed up on the last lines...

Homework Statement The question says that : Find the value of ##a## so that the equation $$x^3-6x^2+11x+a-6=0$$ has exactly three integer solitions. Homework Equations IF ##p##,##q##,##r## are the roots of this equation then: ##p+q+r=6## ##pq+pr+rq=11## ##pqr=6-a## The Attempt at a Solution I...

Homework Statement 2x^2-3x+kx=-1/2 1. k<1 or k>1 2. 1<=k<=5 3. k<=1 or k>=5 1<k<5 Homework Equations b^2-4ac a=2 b=3 c=k The Attempt at a Solution (3)^2-4(2)(k) =9-8k<0 =9/8<k =1&1/8<k I get the answer above but don't know how it relates? Any insight would be appreciated. Thank you, Jaco

Helppp for part (ii). I got 3$e^{\frac{1}{6}\theta i}$

Looking for the general equation for repeated complex conjugate roots in a 4th order Cauchy Euler equation. This is incorrect, but I think it is close: X^alpha [C1 cos(beta lnx) + C2 sin(beta lnx)^2] I think that last term is a little off. Maybe C2 sin [beta (lnx)] lnx ?

Homework Statement Decompose x5 - 1 into the product of 3 polynomials with real coefficients, using roots of unity. Homework Equations As far as I know, for xn = 1 for all n ∈ ℤ, there exist n distinct roots. The Attempt at a Solution [/B] So, let ω = e2πi/5. I can therefore find all the 5th...

Homework Statement Find the roots of z^4+4=0 and use that to factor the expression into quadratic factors with real coefficients. Homework Equations DeMoivre's formula. The Attempt at a Solution I have been able to identify they are \pm 1 \pm i but i have no idea how to factor the...

Hi everyone. I was given a problem in which the roots of a quadratic function were given. Using those roots, I had to write the quadratic function, with integer coeffecients only. The roots were: (-1+ (sqrt -2))/5 and its conjugate. The equation I have so far is: f(x) = 5(x^2) + 2x + (3/5)...

Is it true that $g$ is a primitive root modulo $p$ if and only if $g^{(p-1)/2} \equiv -1 \pmod p$?

hi all... i have problem about square roots for fast calculation, like below sample : is there fast calculation method not commonly/usually ways. it's possible? please, see my picture? thanks in advance.. susanto

How to solve: a1e-k1x+a2e-k2x+...+ane-knx =0 for x? For example in simple case of n=1,2. a1e-k1x+a2e-k2x=0 the solution will be x=In (a1/a2) / [ k1-k2]. But for terms >2 what will be the solution?

let $a>b>c$ be roots of $\sqrt{2015}x^3-4031x^2+2=0$, find $ b(a+c)$.

I don't recall ever doing this but maybe I have. z2 = a = p [cos Ψ + i sin Ψ] = √3 + i*√3 p = √6 Ψ = π/4 Using the formula in the notes, z = 61/4 * exp[i*(π/4 + 2π*k)/2], k = 0, 1.

Let $p,\,q,\,r$ be real numbers such that the roots of the cubic equation $x^3+px^2+qx+r=0$ are all real. Prove that these roots are bounded above by $\dfrac{2\sqrt{p^2-3q}-p}{3}$.

It's the integral of sqrt(x)/(cubed root(x) + 1) I tried regular u substitution but that didn't let me get rid of all the x's. I also just tried long division but that gave me an answer that didn't match with the actual answer to the problem. The actual answer is 6[1/7 x^(7/6) - 1/5 x^(5/6) +...

find the real roots of the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1 $

Within the context of real numbers, the square root function is well-defined; that is, the function ##f## defined by: ##f(x) = \sqrt{x}## Refers to the principal root of any real number x. Is it true that this is not the case when dealing with complex numbers? Does ##\sqrt{z}##, where ##z ∈ ℂ##...

Hi, I have a question that came into my mind while solving some problems. If I have a constant times an expression in a square root like ##4\sqrt{16}## I can square the constant and push it into the square root: ##4\sqrt{16}=\sqrt{4^2 16} = 16##. But what if the constant outside of the square...

i need to create a function that returns the pure zeros on the left semiplane and 0 if there is one and only one zero in the origin of the referential. the return has to be in a column vector like [root1;root2;...;rootn;0] or [root1;root2;...;rootn] if there is no root in the origin of the...

Hello, i'm having trouble evaluating my gamma factor for my special relativity homework, because I need to compute 1 minus a very small number (8.57*10^-13). My calculator treats this value as simply 1, as does Mathematica. Although I don't know much about it, and maybe there's a way to force...

Find the sum of the real roots for $2x^8-9x^7+20x^6-33x^5+46x^4-66x^3+80x^2-72x+32=0$.

I have to show the roots of $$x^{2}-8x-29=0$$ are c$$\pm$$d$$\sqrt{5}$$ I used completing the square method. Once I used CTS I got the answer $$(x-4)^2-45=0$$ So I am not sure what is the next step to put it in the form of c$$\pm$$d$$\sqrt{5}$$

Dear all, please see the page above, (Alan F, Beardon, Abstract Algebra and Geometry). On the page, Theorem 3.5.2 says that the set of Complex numbers from ## z^n = 1 ##, where ## |z| = 1 ## forms a group w.r.t multiplication. I want to know if... The inverse of all elements...

Homework Statement $$f:\mathbb{R} \rightarrow \mathbb{R},$$ $$ f(x) = \frac{1}{\sigma \sqrt{2 \pi}} e^{\frac{-(x-\mu)^2}{2 \sigma ^{2}}}$$ What are the roots of this equation? Homework EquationsThe Attempt at a Solution The roots of an equation are the values of x such that f(x) = 0. This...

Hi, I would like to know how the RPM of a root blower will affect the CFM and pressure created by the blower. The confusion that I have is this: Generally when RPM is decreased, both the CFM and pressure decreases. But according to Bernoulli's principle, should not one decrease and the other...

Homework Statement For every real x>0 and every n>0 there is one and only one positive real y s.t. yn=x Homework Equations 0<y1<y2 ⇒ y1n<y2n E is the set consisting of all positive real numbers t s.t. tn<x t=[x/(x+1)]⇒ 0≤t<1. Therefore tn≤t<x. Thus t∈E and E is non-empty. t>1+x ⇒ tn≥t>x, s.t...

Homework Statement If both roots of the equation ax^2 + x + c - a = 0 are imaginary and c > -1, then: Ans: 3a < 2+4c Homework Equations Discriminant < 0 for img roots Vieta The Attempt at a Solution 1-4(a)(c-a)<0 4ac > 4a^2 + 1 Minimum value of 4a^2 + 1 is 1 so 4ac>1 I can't think of...

Homework Statement Solve: \frac{d^{2}y}{dx^{2}} + \omega^{2}y = 0 Show that the general solution can be written in the form: y(x) = A\sin(\omega x + \alpha) Where A and alpha are arbitrary constants Homework EquationsThe Attempt at a Solution I know that I will need to change variables for...

Let $$\mathbb{Q}(\sqrt{2},\sqrt{3})$$ be the field generated by elements of the form $$a+b\sqrt{2}+c\sqrt{3}$$, where $$a,b,c\in\mathbb{Q}$$. Prove that $$\mathbb{Q}(\sqrt{2},\sqrt{3})$$ is a vector space of dimension 4 over $$\mathbb{Q}$$. Find a basis for $$\mathbb{Q}(\sqrt{2},\sqrt{3})$$. I...

Hi, In the attached image the roots are shown for the characteristic equation. I don't know how the roots were found. Anyone able to help? Thanks Splint

Show that all real roots of the polynomial $f(x)=x^5-10x+38$ are negative. Note: I know this is a fairly easy challenge, but it's good to see how different approaches can be generated from different people so that we can learn from one another. :o (Yes)

I have to find this: $$\lim_{{x}\to{3}}\frac {\sqrt{6x - 14} - \sqrt { x+1}}{x-3}$$ So I do this: $$\lim_{{x}\to{3}}\frac {\sqrt{6x - 14} - \sqrt { x+1}}{x-3} * \frac{\sqrt{6x + 14} + \sqrt{x+1}}{\sqrt{6x + 14} + \sqrt{x+1}}$$ The top part is easy since $$(\sqrt{a} - \sqrt{b})(\sqrt{a} +...

Homework Statement Decide the inverse laplace transform of the problem below: F(s)= \frac{4s-5}{s^2-4s+8} You're allowed to use s shifting. Homework Equations The Attempt at a Solution By looking at the denominator, I see that it might be factorized easily, so I try that...

I'm trying to do some refreshing of differential equations featuring damped systems. Specifically, I have a question regarding the differential equation solution to an under damped system involving complex roots. Referring to the attached pdf, an under damped system will yield a complex...

How do I solve, 1: (√3*√3*√3)/(√3+√3+√3)? How do I simplify it? I'm confused on how to shorten √x+√x+√x, I just don't get it. Also if: √700 = 26.46, then how is √70000 = 264.6? Shouldn't it be 2646?

Let $P(x)=x^2+6x+k$ for all real $x$, where $k$ is some real number. For what values of $k$ does $P(P(x))$ have exactly 3 distinct real roots?

Hello I didn't know in which forum to put this... I solved a linear algebra question, and my answer was: {1}^{1/3} which to my understanding is 1. In the book however, they said it is equal to cis 120k k=0,1,2,... where 120 is degrees. I tried taking the complex number 1+0i and turn it into...