Cubic equations of the form $ax^3+bx^2+cx+d$ can be solved in various ways. Some are easy to easy to factor in a pair, for some the roots can be found out by trial-and-error, some are one-of-a-kind, some can be reduced to a quadratic equation. A compilation of all possible ways to solve cubic equations would be very helpful for students and learners.
Just like if we have any quadratic equation which has complex roots, then we are not able to factorize it easily. So we apply quadratic formula and get the roots.  Similarly if we have a cubic equation which has two complex roots (which we know conjugate of each other) and one fractional root, then we are not able to find its first root by hit &amp; trial.  So my question is like quadratic formula, is there exist any thing like cubic formula which help in solving cubic equations?  For example, I have an equation $$2x^3+9x^2+9x-7=0\tag{1}$$ and I have to find its solution which I am not able to find because it has no integral solution. Its solutions are $\dfrac {1}{2}$, $\dfrac{-5\pm \sqrt{3}i}{2} $, I know these solutions because this equation is generated by myself.  So how can I solve equations like these?  Also while typing this question, I thought about the derivation of quadratic formula, which is derived by completing the square method.  So I tried to apply ‘completing the cube’ method on the general equation $ax^3+bx^2+cx+d=0$ but it didn't help.  So please help me in finding a cubic formula or to solve the equations like given in example by an alternative method.
How could I adjust the size and placement of chapter heading in my custom class? I am using report class as my base. When I used titlesec package inside my custom class, it threw error     ! Package titlesec Error: Not allowed in `easy'   settings. Does a List guarantee that items will be returned in the order they were added? Does a List&lt;T&gt; always guarantee that items will be returned in the order they were added when enumerated?  Updated: Thanks for all the answers folks, puts my mind at ease. I had a quick poke around the List&lt;T&gt; class with .NET Reflector (should've probably done that in the first place) and indeed the underlying store is an array of T (T[]). Why does Ubuntu create a file ending in ~ for every file I create with a text editor like GNU Emacs? In Windows 8, when Alt+Tab, the desktop would appear as an option to switch to at the end of the list of open programs (thus Alt-Shift-Tab helpfully took you right to the desktop).  In Windows 10 that desktop option is gone. Is there any way to get it back?  I know about Windows+D but Alt+Tab is just more convenient in my opinion. And  also has no desktop and is uglier. Detect whether Excel workbook is already open Is there a way to make an Android phone (HTC Desire HD) act as Bluetooth speakers or headset?  I'd like to use the phone as wireless headphones whilst watching a movie on a computer (running Ubuntu). $$\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$$  when f is continuous on $[0,1]$  I know it can be proved using bounded convergence theorem but,  I wanna know proof using only basic properties of riemann integral and fundamental theorem of calculus and MVT for integrals ...   Thank you. what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order.  So far, I have not found any precise definition in literature. Showing equality of sequence of holomorphic functions to limit function if converges uniformly locally I do not know how to proof following question:  Let $D$ be a domain in $C$.  ${{f}_{n}}$ be a sequence of holomorphic functions on $D$ that converges uniformly locally to a function non-constant $f$. Show that for each $z\in D$, there is a sequence ${{z}_{n}}\to z$ as ${{f}_{n}}({{z}_{n}})=f(z)$ for all sufficiently large $n$.  Actually, I know that I should use an assumption such as: ${{f}_{n}}({{z}_{n}})\ne f(z)$  Therefore, there is compact set    that  $\left\| {{f}_{{{n}_{k}}}}({{z}_{{{n}_{k}}}})-f(z) \right\|&gt;\varepsilon $  Then I think I have to use Montel theorem, but I do not know how to show the contradiction. Any examples of life forms on Earth that could have survived on Mars or Moon? Do we know of any living forms on Earth that could continue to live on Mars? There is currently no way to delete questions or answers in the iOS app, even with the required Trusted User privilege. We can edit and vote, but not delete.  Can this option please be added to the app? What are the different ways that a message can be displayed to a bash shell after a user logs in? Goal: I want to be able to securely use the internet via my home PC while my notebook is connected to an open hotspot/access point.  I do know that I can use a SSH tunnel/SOCKS proxy, but I don't want to fiddle around with applications (making them use it, if even possible). I guess what I need is an OpenVPN setup, so I'm looking for a detailed guide on how to:   Install and setup the OpenVPN server Setup the OpenVPN client (NetworkManager)   Ubuntu versions this should work on are 10.10 and 11.04. Are black holes in a sense transparent? Meaning, they don't 'hide' any stars that happen to be 'behind' them?  A regular object obviously blocks a point source of light emanating from 'behind' it. Meaning it creates an area of space photons cannot get to us from. But a massive object curves space, and so via gravitational lensing we can sometimes see distant stars that are 'behind' other stars (by behind I mean that if the star in the middle would suddenly vanish, the trajectory of photons traveling from the distant star to Earth would have passed through the volume now occupied by the middle star).  My thinking, is that black holes curve space so much, that from ANY source of light outside it, light has a trajectory to reach us on Earth (assuming of course no other object blocks it on its way).  The reasoning for this is: Suppose we draw the straight line between us and the light source (as if the black hole wasn't there at all). Photons from our light source crossing the event horizon would obviously 'fall' inside. A photon in theory traveling exactly tangent to the event horizon would in a sense go 'into orbit' around the black hole (obviously there is no such thing, because the black hole isn't perfectly spherical, and there are quantum effects and so forth. But I'm focusing on the geometry), crossing our straight line exactly at the intersection of our line and the event horizon. Any photon passing near the event horizon, but a bit farther, would follow a curve that intersects our line a little farther from the event horizon. Continuing this, there would be at last be a photon that would exactly curve our way. Many of the people have asked almost the same question.  I'm trying to render a big mesh + fur/grass file with a few characters on the screen.  when I hit render, after a number of processing, copying data to gpu etc... i get following error.  Mem: 2960.07M, Peak:2960.07M | CUDA error: Out of memory in cuMemAlloc(&amp;device_pointer, size).  I'm using Geforce GTX 690 which contains 4GB memory.   Can somebody tell me how can i get GPU rendering, while 2960M is far less then 4196M of memory.   I know somebody would suggest me to use CPU.. but i'm curios why not GPU?   I've already compressed texture, etc   Edited  I don't think any other application is using any of my 1GB remaining memory. everything is closed but blender is there, also the GPU utilities are showing 2000/2000 MB are free on one card and 1900/2000MB is free on 2nd card.    i'm using GTX 690. with two GPUs of 2GB each. If somebody thinks that one GPU should have memory matching to the cuda memory/or Peak memory then I'm doubtful. Because in preferences Blender does know that I have 2GPUs where I can select either first GPU or second GPU or even both.   When I render some of smaller objects/scenes both of my GPU renders simultaneously with two blocks.  I'm running following specs  Xeon E5-1660 0 24GB - 1600mhz GTX690 What is the difference between "What are you?" and "Who are you?"? I am often asked "Who are you?" and "What are you?", but I don't know the difference between these two questions.  Please explain it to me. Is the vector cross product only defined for 3D? Fibonacci Sequence and series limits Relation between magnetic moment and angular momentum -- classic theory How do I prove the relation between the vectors of magnetic moment $\vec\mu$ and angular momentum $\vec L$, $$\vec\mu=\gamma\vec L$$  ?  Many text books and lecture notes about the principles of magnetism show the relation of $\mu$ and $L$ as scalars only and then just state that the relation holds also for the vectors. An example:   $$\mu=I\cdot A = \frac{q}{t}\pi r^2 = \frac{qv}{2\pi rm}m\pi r^2$$  ($I=q/t$: current, $A$: area of a loop, $q$ charge, $t=2\pi r/v$: time of 1 rotation, $v$: velocity of particle, $m$: mass )  The angular momentum is $\vec L = \vec r\times\vec p$ or $L=mrv$ and therefore $$\Rightarrow \mu = {\frac{q}{2m}} L = \gamma L.$$  Why is this also true for the vectors? Is there a general explication by classical physics without the need of quantum theory? How do I have a camera follow my object in Unity? I have an object that automatically moves by itself and I want the main camera to automatically follow it.(Like in games such as geometry dash and jetpack joyride)  This is the code for the automatic moving object in case it is needed:  using UnityEngine; using System.Collections; public class automove : MonoBehaviour { public static int movespeed = 5; public Vector3 userDirection = Vector3.right;    public void Update() {     transform.Translate(userDirection * movespeed * Time.deltaTime);  } }   So does anyone know any good scripts I could add to the main camera to follow this object that automatically moves? Thanks in advance!