The quadratic formula can be used to find the roots of any quadratic polynomial of the form $ax^2 + bx + c$:  $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$  The derivation is simple enough and uses a technique called completing the square.  Is there a formula to solve cubic equations of the form:  $$ax^3 + bx^2 + cx + d$$  Or a general formula to solve an $n$th degree polynomial?  EDIT: According to the answers below, and some mathematicians, such a formula isn't possible. But why is that? I mean, consider an $n$th degree polynomial $p(x)$ with roots $r_1, r_2, \dots, r_n$:  $$p(x) = ax^n - \left(\sum r_i\right)ax^{n-1} + \left(\sum_{i, j} r_ir_j\right)ax^{n-2} + \dots+(-1)^n(r_1r_2\dots r_n)$$  We have $n$ equations with $n$ unknowns, surely this can be solved using a general algorithm?
I know that there is a precise solution of the equations:  \begin{align}     x + a &amp; = 0 \\     x^2 + a\cdot x + b &amp; = 0 \\     x^3 + a\cdot x^2 + b\cdot x + c &amp; = 0 \\     x^4 + a\cdot x^3 + b\cdot x^2 + c\cdot x + d &amp; = 0 \end{align}  Why there is no solutions for equation of degree $\ge 5$?
A calligraphic font with small (lowercase) letters Is there a widely available calligraphic font that (1) contains small (lowercase) letters and (2) has a recognizably calligraphic capital letter O?  Background. (1) In mathematics (algebraic geometry), , the latter being written in calligraphic script (standard notation for sheaves). However, most books end up typesetting the sheaf as follows    pulling the small letters from the italics of a completely different font.  (2) The calligraphic capital letter O plays a special role in algebraic geometry, too, and most authors insist on it being recognizably calligraphic (i.e. not just a closed round loop)    Edit. The following are achieved with extra kerning from @egreg's and @UlrikeFischer's answers (in chronological order)    @egreg's answer  \makeatletter \DeclareFontEncoding{LS1}{}{} \DeclareFontSubstitution{LS1}{stix}{m}{n} \DeclareMathAlphabet{\mathcal}{LS1}{stixscr}{m}{n} \makeatother  \DeclareMathOperator{\sHom}{\mathcal{H\mkern-7mu o\mkern-2.5mu m\mkern-1.5mu}} \DeclareMathOperator{\sExt}{\mathcal{E\mkern-4.5mu x\mkern-2.5mu t\mkern-1mu}} \DeclareMathOperator{\sEnd}{\mathcal{E\mkern-4mu n\mkern-4.5mu d\mkern-1mu}}       @UlrikeFischer's answer  \DeclareFontFamily{U}{mathc}{} \DeclareFontShape{U}{mathc}{m}{it}% {&lt;-&gt;s*[1.03] mathc10}{} \DeclareMathAlphabet{\mathcal}{U}{mathc}{m}{it}  \DeclareMathOperator{\sHom}{\mathcal{H\mkern-3mu om}} \DeclareMathOperator{\sExt}{\mathcal{E\mkern-3mu xt}} \DeclareMathOperator{\sEnd}{\mathcal{E\mkern-3mu nd}} How can I listen to communication between a process and a serial port? On a Gentoo Linux machine, a process is communicating with another computer via serial port (/dev/ttyS1). I want to listen to this communication (i.e. what is being sent/received by the process) without further interfering.   How can I achieve this with just basic Linux tools? I even can change the serial port the process is communicating with, so maybe I can 'shortcut' two devices (/dev/ttyS1 and /dev/xxx) listen to all communication, when all output to /dev/ttyS1 is forwarded to /dev/xxx and vice versa. But how do I do it? Something with socat? The new software center in Ubuntu 16.04 shows no application data found This is the error I get when I open the software center. I've tried rebooting and running sudo apt-get update but I see no improvement. It worked earlier but I don't know what went wrong since then. I'm using Ubuntu 16.04 final release on a Dell Inspiron 15R. Any advice will be appreciated. Thanks in advance.  EDIT: The problem magically solved itself without me doing anything special (or maybe inadvertently doing something, I don't know). So I'd still like to get to the root of this. OpenVPN restriction Ive set up an OpenVPN server at my place and handed out a certificate to a client who has an openvpn client at his place.  I need require that he should be able to connect to the vpn server only from one IP address (his current one).  How can I do this? tmux: shared session, one user in a pane, another in another pane, two different cursors I've decided to try tmux: have been reading the docs and googling around, trying to find a way to have two users sharing a session, each with a different cursor.  However, giving 777 permissions to the socket, or creating a group, chgrping the socket and adding both users to it, seems to let that same socket be used to share a session with only one cursor: both users can write, but always in the same cursor position.  Right now both users are in the same home server over ssh, and the idea is to be able to have:   A terminal in a, let's say, left pane, where I can type commands Another terminal in a right pane, where I can see another user typing commands  in his own terminal The same thing for the other user   What I'm doing at the moment is using two sessions (not shared) and a script -f and tail -f combination that kinda works for reading each other's key strokes, but I reckon there is probably some way of doing this using tmux sharing capabilities.  Is there a way to get this idea working with write support in each other's terminal?  What is the better way to do this? How would one prove that $$\lim_{n\rightarrow \infty} \frac{F_{n+1}}{F_n}=\frac{\sqrt{5}+1}{2}=\varphi$$  where $F_n$ is the nth Fibonacci number and $\varphi$ is the Golden Ratio? Should an image carousel on a mobile site automatically rotate? Or should the user have complete control? The intention is to have a slider/carousel that shows 6 or 7 images which the user can look through by clicking arrow buttons left or right. But the question has also come up if the user has not started looking through the images, should the carousel start automatically rotating the images. Since this is for a mobile site, I'm uncertain... Pure significance of line integrals of vector fields I can understand how the line integral of a scalar function has pure-mathematical significance, representing an area beneath a curve. But is there a pure significance to the line integral of a vector field? The only understanding I have of this is the physical example of work done on a moving particle. Thanks for any insight. Changing left margin in itemize environment of beamer class I can't figure out how to line up the symbols of an itemize environment in beamer with the left part of the main text. In other words, I would like to supress the left margin in beamer itemize lists.  It would be nice if the solution worked with any itemize label defined with  \setbeamertemplate{itemize item}[]   In order to make myself clearer, what beamer produces by default is something like:  Main text. Main text. Main text...    * Item 1    * Item 2    * Item 3    ....   and the desired result is   Main text. Main text. Main text... * Item 1 * Item 2 * Item 3 .... How discontinuous can a derivative be? There is a well-known result in elementary analysis due to Darboux which says if $f$ is a differentiable function then $f'$ satisfies the intermediate value property.  To my knowledge, not many "highly" discontinuous Darboux functions are known--the only one I am aware of being the Conway base 13 function--and few (none?) of these are derivatives of differentiable functions.  In fact they generally cannot be since an application of Baire's theorem gives that the set of continuity points of the derivative is dense $G_\delta$.  Is it known how sharp that last result is?  Are there known Darboux functions which are derivatives and are discontinuous on "large" sets in some appropriate sense? I am having an error compiling the following code with xelatex and lualatex:  ** WARNING ** "seac" character deprecated in Type 2 charstring. ** ERROR ** Type2 Charstring Parser: Parsing charstring failed: (status=-1, stack=5)  Output file removed.  \documentclass{article}  \usepackage{xunicode} \usepackage{fontspec}  \setmainfont[Mapping=tex-text,Ligatures={Common}, Numbers={OldStyle}]{IM FELL English PRO} \begin{document}  Buongiorno a tutti. Trallallà.  \end{document}   The error disappears when I omit the accented à. It also doesn't happen when I use other fonts.  Could you suggest any workaround?  Thank you, A When would you use a WeakHashMap or a WeakReference? The use of weak references is something that I've never seen an implementation of so I'm trying to figure out what the use case for them is and how the implementation would work. When have you needed to use a WeakHashMap or WeakReference and how was it used? I started learning Bash a couple of days ago.  I'm trying to obtain an exit status of grep expression into a variable like this:  check=grep -ci 'text' file.sh   and the output that I got is   No command '-ic' found   Should I do it with a pipe command? Get attention to a new answer to an old question It happened to me that I found a solution to an older question. This question was commented by many people with comments like "very interesting, important, ..." such that I thought that my answer has also some value. My answer was the only one. It was upvoted and accepted by the asker. However, I assume, that very few of the "commentators" saw my answer, because the system does not inform automatically about new answers.   I think it would have been helpful for me and the commentators when they would get informed. I could achieve this by adding my own comment to their comments, because this triggers a system message. My question is: would this action be considered admissible or not? Are there some other possibilities? Prove that for any infinite poset there is an infinite subset which is either linearly ordered or antichain. Prove that any finite group $G$ of even order contains an element of order $2.$ change color of pgfplot bar plot I want to plot two bar plots next to each other. This works without a problem. However both use the first color defined in a pre or selfdefined colorseries. I tried to modify the color manually in the plot, but the color and fill definitions have no effect:  \documentclass[border=5mm]{standalone} \usepackage{pgfplots, pgfplotstable}  \begin{document}  \pgfplotstableread[header=false, col sep=comma]{  1, first item 5, second item }\datatable  \begin{tikzpicture}   \begin{axis}[             ,xbar     ,bar width=2ex, y=3ex     ,enlarge y limits={abs=0.75}     ,ytick={data} % (\empty,data,{coordinate list})     ,yticklabels from table={\datatable}{1}   ]   \addplot table [     ,red!20!black,fill=red!80!white,     y expr=-\coordindex, % Use negative coordinate index as y coordinate     x index=0 % Use first column as x coordinate   ] {\datatable};   \end{axis} \end{tikzpicture}  \end{document}   The plot is still in the default blue. Frequently when I ask a question I have to include quite a bit of code for reference purposes.    It would be nice to be able to bold the important lines in the code.  For example this code:  &lt;DataTrigger Value="True"&gt;     &lt;DataTrigger.Binding&gt;         &lt;MultiBinding Converter="{StaticResource DisableWorkItemConverter}"&gt;             &lt;Binding ElementName="MainForm" Path="PickedWorkItemID"/&gt;             &lt;Binding Path="Id"/&gt;         &lt;/MultiBinding&gt;     &lt;/DataTrigger.Binding&gt;     &lt;Setter Property="IsEnabled" Value="False"/&gt;     &lt;Setter Property="IsSelected" Value="False"/&gt; &lt;/DataTrigger&gt;   Is not way long, but the important part (as seen in ) is the second to last line.  However, if I try to bold it, it ends up looking like this:  &lt;DataTrigger Value="True"&gt;     &lt;DataTrigger.Binding&gt;         &lt;MultiBinding Converter="{StaticResource DisableWorkItemConverter}"&gt;             &lt;Binding ElementName="MainForm" Path="PickedWorkItemID"/&gt;             &lt;Binding Path="Id"/&gt;         &lt;/MultiBinding&gt;     &lt;/DataTrigger.Binding&gt;     &lt;Setter Property="IsEnabled" Value="False"/&gt;     **&lt;Setter Property="IsSelected" Value="False"/&gt;** &lt;/DataTrigger&gt;   Not what I was going for.  It would be nice to be able to draw attention to specific lines of code when needed.    (One suggestion of a different color background sounds nice too (ie a yellow background to draw the eye to the relevant code.)  If there is some other way to bold in code please let me know. out file with multiple criteria in python? When I'm watching a film in Mythtv the screen turns to black every 10 - 15 mins and I have to log back into Ubuntu. Very annoying!  How do I disable the black screen / screensaver / logout in Unity?  There no longer seems to be any options to turn the screen saver off as there were in Ubuntu prior to Unity.