Root Definition and 918 Threads

Proof: Consider the equation of ## a^2-x^2=\epsilon\sinh x ## for ## 0<\epsilon<<1 ##. Let ## \epsilon=0 ##. Then the unperturbed equation is ## a^2-x^2=0 ##. This gives ## a^2-x^2=0\implies (a+x)(a-x)=0\implies x=\pm a ## with the root ## x=x_{0} ## such that ## x_{0}(a)=a ## because ## x\geq...

By IVT and trial and error, I get the interval to be ##(-\frac{1}{2},-\frac{1}{4})## I don't know how to do the next part. Let the actual root of the polynomial be ##x_{0}## and the approximate value is ##p##, we have ##|p-x_{0}|<\frac{1}{8}## I am not sure how to continue. Thanks

I can't find a short solution without using calculator.

Many people have said that the noise that affects laser light is proportional to the square root of the illumination. But I can't find the formula. Can anyone help?

Hmmmm was a nice one... took me some time to figure out ...seeking alternative ways ... My working; ##x=\dfrac{-2±\sqrt{4+4 \sinh^2 2u}}{2 \sinh 2u}## ##x=\dfrac{-2±2\sqrt{1+ \sinh^2 2u}}{2 \sinh 2u}## ##x=\dfrac{-1±\sqrt{1+ \sinh^2 2u}}{\sinh 2u}## ##x=\dfrac{-1+\sqrt{1+ \sinh^2 2u}}{\sinh...

I have used root locus before but my confusion now is that the input is the negative feedback. Usually when I have negative feedback I consider the the error between the input (ideal) signal and the observed signal. Also, in this case what is the tranfer function since u = -k*y, and what does...

Hi, is it possible, is there any formula that can help me to take root from (for example) 1,2 without a calculator (by hand)? For example, there is a cos(x) formula that can be calculated on the paper: $$\cos x=\sum_{n=0}^{\infty}(-1)^{n} \frac{x^{2 n}}{(2 n) !}$$ There is the Babylonian method...

The correct answer is; ##\sqrt{\dfrac{16}{64}}=\dfrac{4}{8}## . I do not seem to understand why some go ahead to simplify ##\dfrac{4}{8}## and getting ##\dfrac{1}{2}## which is clearly wrong. I do not know if any of you are experiencing this... I guess more emphasis on my part. Cheers! Your...

I feel incredibly stupid for not getting this. I found this math problem in the beginning of my precalculus book: 12√x^4 That's 12th root of x to the fourth power. How do I find the root of x if the root is larger than the exponent?

When you use the power rule to differentiate the square root, the result is 1/2(sqrt. x) which is undefined at 0. But, when you use the definition of the definition of the derivative to calculate it, the result is infinity. What causes this difference between these two methods?

Is it always true that the square root of an odd powered integer will always be irrational?

##\sqrt{3}## is irrational. The negation of the statement is that ##\sqrt{3}## is rational. ##\sqrt{3}## is rational if there exist nonzero integers ##a## and ##b## such that ##\frac{a}{b}=\sqrt 3##. The fundamental theorem of arithmetic states that every integer is representable uniquely as a...

Proof: We will first show ##\gcd(p(x), p'(x)) = 1##. Define ##d(x) = \gcd(p(x), p'(x))##. Then we can find ##q(x) \in F[x]## such that ##p(x) = d(x)q(x)##. But ##p(x)## is irreducible which means ##d(x)## is constant or ##q(x)## is constant. If ##q(x)## is constant, then ##\deg d(x) = \deg...

Is the odd root of an even number always an irrational number? For example the 7th root or the 11th root, etc. of an even number.

AIUI, an algebraic is defined as a number that can be the solution (root) of some integer polynomial, and is any number that can be constructed via any binary arithmetic operation or unary root operation with arguments that are themselves algebraic numbers. I have been able to prove this for...

Hi I was working on a physics problem and it was almost solved. Only the part that is mostly mathematical remains, and no matter how hard I tried, I could not solve it. I hope you can help me. This is the equation I came up with and I wanted to prove it: $$\lim_{n \rightarrow+ \infty} {...

in order to solve a series, the root test is applied and I have this limit ## \lim_{n \rightarrow +\infty} \sqrt[n] {\left| {\frac {i^{\frac { n} {3}}} { \frac {2n} {3} +1}} \right| } ## I don't understand why at the second step the numerator becomes 1, cannot recall why it becomes 1, that is...

In https://mathworld.wolfram.com/InnerProduct.html, it states "Every inner product space is a metric space. The metric is given by g(v,w)= <v-w,v-w>." In https://en.wikipedia.org/wiki/Inner_product_space , on the other hand, "As for every normed vector space, an inner product space is a metric...

$\tiny{GRE.al.06}$ For the polynomial $x^3-3x^2-6x+8\quad -2$ is the smallest root. Find the largest root. $a.\, -1 \quad b.\, 1 \quad c.\, 2 \quad d.\, 3 \quad e.\, 4$ Since -2 is a root then use synthetic division $\begin{array}{r|rrrr} -2&1&-3&-6&8\\ & & -2& 10&-8\\ \hline &1& -5&...

My attempt: \begin{align} \lim\limits_{n \to \infty} \sqrt{n^2 + n} - n &= n\sqrt{1+\frac{1}{n}} -n\\ &=n - n\\ &= 0\\ \end{align} I think the issue is at (1)-(2) For comparison, here is Rudin's solution

Summary:: solution of first order derivatives we had in the class a first order derivative equation: ##\frac{dR(t)}{dt}=-\sqrt{\frac{2GM(R)}{R}}## in which R dependent of time. and I don't understand why the solution to this equation is...

I have to solve a certain numerical problem without using calculator and furthermore, there is a time limit for solving this problem. The answer I have got so far is ## \sqrt{\frac{100}{99}}## I know I can reduce the numerator to 10 but then I am stuck with square root of denominator which is...

I ran into an interesting video on Youtube yesterday, about a fast way to compute the reciprocal of the square root of a number. I.e., to compute this function: ##f(x)= \frac 1 {\sqrt x}## The presenter is John Carmack. If you search for "Fast Inverse Square Root — A Quake III Algorithm" you'll...

Hi I am trying to install all the necessary files for VS to run games, I just have a very simple question. This is the instruction. When the download completes, create a folder at the root of the same drive where you installed Visual Studio and name it SFML. Also, create another folder at the...

I consider three cases, based on the sign of ##a_0##. if ##a_0 == 0##: Set ##x=0##. \begin{align*} f(0)&=&a_0+a_1\cdot 0+a_2\cdot 0^2+a_3\cdot0^3+a_4\cdot0^4+0^5\\ &=&a_0+0\\ &=&0+0\\ &=&0 \end{align*} elif ##a_0<0##: Define ##M=\max\{|a_i|:1\leq a_i\leq 5\}## and set ##x=5(M+1)\neq 0##...

Prove that the polynomial $P(x)=x^{13}+x^7-x-1$ has only one positive zero.

I started out by rewriting the function as (f(x^2))^(1/2). I then did chain rule to get 1/2(f(x^2))^(-1/2) *(f'(x^2). - I think I need to go further because it is an x^2 in the function, but not sure.

I wanted to install MySQL on my laptop running Ubuntu 20.04, and was following this website for instructions. I executed the following commands: ~$ sudo apt update ~$ sudo apt install mysql-server ~$ sudo mysql_secure_installation After installation, I found that I could log into root user...

ROOT is required as a pre-requisite for some software that I am trying to install. I'm on a MacOs system and I have tried to install using 'brew install root'. Do I need to do anything else? How can I check that root was successfully installed? When I tried to install said software, apparently...

Let $a$ and $b$ be real numbers such that $a\ne 0$. Prove that not all the roots of $ax^4+bx^3+x^2+x+1=0$ can be real.

Summary:: Why are you multiplying by 1000NM/kJ within square root? Practice problem for FE [Thread moved from the technical forums so no Homework Template is shown]

if x_{I}, I = {1,2,...,2019} is a root of P(x) = ##x^{2019} +2019x - 1## Find the value of ##\sum_{1}^{2019}\frac{1}{1-\frac{1}{X_{I}}}## I am really confused: This polynomial jut have one root, and this root is x such that 0 < x < 1, so that each terms in the polynomial is negative. But the...

Let us suppose I have a number ##x## such that ##x<0##. If I want to write the roots of the ##x^{1/n}##. How can we write the roots of this number. I thought we can write $$|x|^{1/n}e^{i\pi\theta}$$ for ##\theta = \frac{2l + 1}{n}## and ##l = 0,1,2## etc. Is this correct ? Similary If I...

This is a quote from "Calculus", by Robert A. Adams. It's a translation from spanish: "Roots of square numbers If ##a## is a positive real number, there exist two different real numbers whose square is ##a##. They are ##\sqrt{a}\;## (the positive square root of ##a##) ##-\sqrt{a}\;## (the...

Hey! 😊 We have the following iteration from Newton's method \begin{align*}x_{k+1}&=x_k-\frac{f(x_k)}{f'(x_k)}=x_k-\frac{x_k^n-a}{nx_k^{n-1}}=\frac{x_k\cdot nx_k^{n-1}-\left (x_k^n-a\right )}{nx_k^{n-1}}=\frac{ nx_k^{n}-x_k^n+a}{nx_k^{n-1}}\\ & =\frac{ (n-1)x_k^{n}+a}{nx_k^{n-1}}\end{align*} I...

Firstly, the cube root of 17 is 2.571281591 which is 2.57 to 3.s.f. Initially, I thought about just approaching this problem using the Newton-Raphson Method when x0=2. In which case; x^3=17 x^3-17=0 Using the Newton-Raphson iterative formula xn+1=xr-f(xn)/f’(xn) f(x)=x^3-17 f’(x)=3x^2...

For the first question, i believe that mechanical energy is conserved hence we can derive the total energy i think. In regards to the second question, I'm assuming its at room temperature, so helium is monotonic therefore it has 3 degrees of freedom, therefore its internal energy is 3/2KT. I am...

find the square root of ## a+b+√(2ab +b^2)## let ##√[a+b+√(2ab +b^2)]= ±(√x +√y)## then, ##a+b+√(2ab +b^2)= x+y+ 2√(xy)## where ##a+b=x+y##.......1 ##(b+a)^2-a^2=4xy## .....2 from 2, ##a^2=(b+a)^2-4xy## ##a=√[x+y)^2-4xy]## ##a=√[x^2-2xy+y^2]## ##a=x-y## therefore...

Please Help me solve it \[ \sqrt{x}+\sqrt{x+8}=8 \] thanks

This question came in NEET Exam 2018.Now my first query is that in the question,the mass of one Oxygen molecule is given wrong.Its exactly half it's true value.I don't think anybody has noticed this before because I couldn't find any change in the printed question on so many different books...

I want to find the analytical solution to the integral given below. \int_{-\infty}^{\infty} \frac{ sinc^2(\frac{k_yb}{2})}{\sqrt{k^2 - k_x^2 - k_y^2}}dk_y In other words, \int_{-\infty}^{\infty} \frac{ \sin^2(\frac{k_yb}{2})}{(\frac{k_yb}{2})^2\sqrt{k^2 - k_x^2 - k_y^2}}dk_y Can this be...

Could it be said that since ##a=A(f(x))\sqrt{f(x)}##, with ##A(x)\in\{1,-1\}## then ##a^2=f(x)##,, that ##a## is the square root of ##f(x)## ? In other words could the sign of the root depend on the argument inside it ? Else it would have to be chosen by human free will and to be blocked for...

As some simple Lie groups and their algebras are essential for our current understanding of QM, I wondered if especially the highest positive (or likewise lowest negative) root can be explained physically. The roots are the weights of the adjoint representation. Are their physical meanings...

Show $20a^2+20b^2+5c^2\ge 64$ if $y=x^4+ax^3+bx^2+cx+4$ has a real root.

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I have yet another question regarding Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need further help with other aspects of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ... I need help with an aspect of Example 1.5, Section 1.2, Chapter III ... Example 1.5, Section 1.2, Chapter III, reads as...

I am reading Theodore W. Gamelin's book: "Complex Analysis" ... I am focused on Chapter 1: The Complex Plane and Elementary Functions ... I am currently reading Chapter 1, Section 4: The Square and Square Root Functions ... and need some help in verifying a remark by Gamelin ... ... The...

On the right paragraph it says "The trial divisor 1200 goes into the dividend 13952, 8 times" Clearly 1200 goes into 13952, 11 times. I don't understand why 8 is (arbitrarily?) chosen. Please help. Thanks.

Finally, after a lot of hesitation, I rooted my phone (Samsung Galaxy On7, Android 6.0.1) using Magisk systemless root. The main aim was to remove pre-installed useless software (often termed "bloatware"). I didn't want to flash Magisk using TWRP, because installing TWRP would remove the stock...