We know that when a quadratic equation $ ax^2 + bx + c = 0 $ has zero discriminant value, then the quadratic equation has only &quot;one root&quot;. But why do some mathematician call it &quot;double root&quot; and also that the equation still has two roots, but their roots are double. When we draw the equation in the coordinate plane, we can see that the graph of the equation will just touch the $ x $-axis at only one point. The second question is: We know that the sum of the roots of a quadratic equation $ ax^2 + bx + c = 0 $ is $ -\dfrac{b}{a} $. But when the discriminant of that equation is zero, the sum is also $ -\dfrac{b}{a} $. It doesn't make sense. For example, if we have quadratic equation $ x^2 + 4x + 4 = 0 $, we can factor it as $ \left(x + 2\right)^2 = 0 $ and so $ x = -2 $. Thus, the sum is just $ -2 $. But with &quot;sum of the roots&quot; formula, the sum is $ -4 $. Which one is correct? Is this the reason we call it double root? Thanks in advance :)
According to the definition on , a multiple  is a root with  $n\geq 2$. For example, in the equation $(x-1)^2 = 0$, $1$ it the multiple (double) root.  Question: Why is $x = 1$ the multiple root of $(x-1)^2 = 0$?  What does multiplicity mean in this case? According to Wolfram multiplicity is the number of values for which a condition holds, or the number of times for which a polynomial equation has a root at a given point. In what way does the polynomial $x^2 - 2x + 1 = 0$ have multiple roots?
Combining Multiple Lists Book series with college students finding a way to open portals to other planets I got the storyline of this book series (I believe it was a series) told to me by a colleague at a former workplace but I can't remember the name of the author or of any of the books in the series. The first book is about a couple of college students who find a way to relatively easily open portals to pretty much wherever they want to go. One day, NASA(?) finally succeeds in sending astronauts to Mars(?). When they have just left the spaceship, one of them suddenly gets a tap on the shoulder, and when he/she turns around, these college students stand there in their homemade space suits, chuckling. I believe these students later also start a company that helps other organizations establish themselves on other planets, or perhaps even &quot;conquers&quot; and sells planets to other organizations. A later book in the series also describes a system that lets people upload their minds to the cloud. Some people choose to live solely in the cloud as virtual creatures, only deciding to &quot;descend&quot; into a &quot;flesh&quot; body when they need to perform some task that requires them to be in physical form. Some people also choose to live in physical form but as several entities, or clones, and sync their mind between the clones through this system, basically allowing them to, for example, run a one-man company with several &quot;instances&quot; of the same human who all share their knowledge and all know exactly what all other clones know in the same moment, effectively creating a super-efficient team of workers. I recently received an invitation to a conference and I'm trying to determine how reputable it is. Is there a good way to go about this? Motivation behind standard deviation? Relationship between complex number and vectors How to choose a lens for my first DSLR? I am considering buying my first DSLR.  How do I know whether I need a 18-200 lens, 17-85, or 18-55? Or something else? Limit with multiple variables I need to evaluate the limit $$\lim_{n\rightarrow \infty}(n-\sqrt{n+\alpha}\sqrt{n+\beta})$$  My work: $$(n-\sqrt{n+\alpha}\sqrt{n+\beta})=n-\sqrt{n^2+\alpha n+\beta n+\alpha\beta}$$ I understand that both $\alpha$ and $\beta$ are just constants. But, now how should I factor this? I understand the calculus behind limits, but the algebra troubles me. How can I take browser screenshots at a higher resolution than my browser supports? How do you remove the ability for a user to make a comment or post on a page? I am building a new website on Wordpress, and all my pages have the add comment box at the bottom. I wish fo rthis to be removed from pages such as Home, Contact Us, About etc etc, but want it to remain on the blogs and galleries.  I have read  but do not really understand what this means to do. is there an easier way to do it. I do not have  under Settings -> Discussions. How to efficiently count the number of keys/properties of an object in JavaScript? Yu Yuphoria bricked after installing custom recovery and boot file My YU YUPHORIA got bricked while I was working with it. I want to install custom ROM into it for that I am installing custom recovery and boot file but the file that is found on Internet was not working and that's why I am stuck with company logo. I can't go into recovery mod, neither it boot, and USB debugging mode is turned off. I tried to install fresh ROM from fastboot but after installing every driver properly command prompt is showing "waiting for the device". Please sugest some way to get rid of this problem. How to check if a column exists in a SQL Server table? I need to add a specific column if it does not exist. I have something like the following, but it always returns false:  IF EXISTS(SELECT *           FROM   INFORMATION_SCHEMA.COLUMNS           WHERE  TABLE_NAME = 'myTableName'                  AND COLUMN_NAME = 'myColumnName')    How can I check if a column exists in a table of the SQL Server database? I tried installing Office 2007 in Ubuntu with Wine but it was throwing an error after installing and also when I tried to open a PowerPoint document and save it. Has anybody gotten Office 2007 installed without any issues? Please help me. how to install packages with apt-get on a system connected via proxy? I've got an Ubuntu 11.10 system connected via a proxy to the internet, and I would like to install packages via apt-get. When I try to do that I get this error message:  sudo apt-get install libboost-program-options-dev [...] Err http://gb.archive.ubuntu.com/ubuntu/ oneiric/main libboost1.46-dev i386 1.46.1-5ubuntu2 407  Proxy Authentication Required   Any ideas? Is there an example of a ring $S$ with identity $1_S$ containing a non-trivial subring $R$ which itself has an identity $1_R$, but $1_R\neq 1_S$ (or equivalently $1_S\notin R$). I'd also like to know under what conditions the identities have to be equal. I know they must be equal if $S$ has no zero divisors, since for every $r\in R$ we have $(1_S-1_R)r=0$  In other words: Is the category of unital rings with identity-preserving morphisms a full subcategory of the category of rings (where morphism are only required to respect addition and multiplication)? I want to construct a complex power series with the radius of convergence $R=1$ that diverges in:  1) $k$ points on the circle $\{|z|=1\}$.  2) countable number of points on the circle  I have already crafted series that converges on the whole circle and diverges on the whole circle.  Can anybody help with the problem?  Update: I have googled a series $\sum{\frac{z^{kn}}{kn}}$ that is an answer for 1). Though I still need help for the countable case. The result of ls * , ls ** and ls *** I know using the command ls will list all the directories. But what does the ls * command do ? I used it and it just lists the directories. Does the star in front of ls mean how deep it will list the directories? Beamer item balls go missing while compiling in XeLateX since yesterday's TeXlive update, however, the same code works with pdfLaTeX. Any solution, please. Here is my MWE: \documentclass{beamer}  \setbeamertemplate{itemize item}[ball] \setbeamertemplate{enumerate item}[ball]  \begin{document}  \begin{frame} \frametitle{A tested title} text \begin{enumerate}     \item this is an item \end{enumerate} text3 \begin{itemize}     \item this is an item \end{itemize} \end{frame} \end{document} Whilst I understand how the RSA algorithm works, I don't understand how the CPU operates when it needs to use the mod function with a huge number $n$, for example.  $n = 8192$ bits; $c = m^e \mod n$;  In essence my question is how does the CPU deal with this huge modulus? least common multiple of $\{1,2,...,n\}$ is bigger than $2^{n-1}$ The least common multiple of $\{1,2,...,n\}$ is greater than $2^{n-1}$ for any $n \ge 3$.   I found this in a MATHEMATICA book, but I don't know how to prove this. Can you help me?  [Edit:  has a discussion of an asymptotic stronger result, but that relies on the . What else is known about this? JL]