The solution to the quadratics is given by  $r = -\dfrac{b}{2a}\pm\sqrt{\dfrac{b^2-4ac}{4a^2}}$, which is shortened to $r = -\dfrac{b}{2a}\pm\dfrac{\sqrt{b^2-4ac}}{2a}$, but I'm wondering how if this is justified, given that $4a^2$ can be negative if $a \in \mathbb{C}$, and $\sqrt{\dfrac{x}{y}} \neq \dfrac{\sqrt{x}}{\sqrt{y}}$ if $x$ and $y$ are negative, but given that we have $a^2$, is this justified?  Is $\sqrt{x/y^2} = \sqrt{x}/y$? Is $r = -\dfrac{b}{2a}\pm\dfrac{\sqrt{b^2-4ac}}{2a}$ always true?
When deriving the quadratic equation as shown in the  () the main proof contains the step: $$ \left(x+{\frac {b}{2a}}\right)^{2}={\frac {b^{2}-4ac}{4a^{2}}} $$ the square root is taken from both sides, so why is  $$\sqrt{4a^2} = 2a$$  in the denominator and not $$ \sqrt{4a^2} = 2\left |a  \right | $$ Could somebody explain this to me? Thank you very much
Proving that $\lim_{x \to 0} \frac{f(x)}{|x|}=1$ implies $f$ is not differentiable at $0$. How can I avoid SSH's host verification for known hosts? How can I share a directory with an another user? Is there a Linux equivalent of Windows' "resource files"? Yesterday, Asaf Karagila's  sparked an extensive discussion on ways of proving that all countable ordinals are embeddable in $\Bbb Q$, and whether particular solutions to this use the Axiom of Choice.  Since the arguments used to prove this can be shady on their use of Choice when formulated tersely, fully fleshed out answers are appreciated.  To be concrete, this question asks for proofs of the following fact:     All countable ordinals $\alpha &lt; \omega_1$ are order-embeddable in the rationals $\Bbb Q$.   Especially answers avoiding the Axiom of Choice are appreciated.    Of relevance and related interest is  (but it's not a duplicate, since that question asks for explicit embeddings). argument about fallacy of diff(M) being a gauge group for general relativity I want to outline a solid argument (or bulletpoints) to show how weak is the idea of diff(M) being the gauge group of general relativity.  basically i have these points that in my view are very solid but i want to understand if there are misconceptions on my part that i'm simply not getting and if its so, i ask for help to make the case more solid, or understanding why it doesn't apply (to gravity):    (thanks to Raymond Streater for making that point completely clear) gauge invariance in electrodynamics is an observation that physical observables are unchanged after a gauge transformation without changing coordinate frame ( we are ask to believe that in gravity someone did the same? that is, someone made the observation that physical observables are unchanged after a diffeomorphism-gauge-transformation, only to later argue that because of this, that there are no physical observables to begin with, that doesn't make a lot of sense, to not say that its just plain stupid circular argument ) . The invariance is of course broken when the theory is quantized and $\hbar$ makes an appearance, because it assumes a preferred scale for certain energies. The key point here is: classical gravity is not special regarding having diff(M) as a symmetry group from bulletpoints 2 and 3, if i cannot infer that Diff(M) is a gauge-invariance of electrodynamics, the same should apply to gravity   For this question, i would say that a valid answer would either disprove any of the arguments as fallacies themselves (hence showing a solid argument why gravity is special and diff(M) is without a doubt its gauge group), or improve the argument for making it bullet-proof (sorry for the pun) Let $p$ a cut point of a connected space $(X, \mathcal{T})$ and suppose $C$ and $D$ form a separation  of $X-\{p\}$, i.e $$X-\{p\}=C \cup D$$ Prove that $C \cup \{p\}$ is conected. I have this idea: Since $p$ is a cut point then $X-\{ p \}$ is not connected, and since $C$ and $D$ form a separation of $X- \{p \}$ then $X-\{p \} = C \cup D$, and also $C \cup D$ is not connected. And also $$C \cap \overline{D}= \overline{C} \cap D= \emptyset$$ I need an idea to continue please How do you debug PHP scripts? Try to solve this puzzle:     The first expedition to Mars found only the ruins of a civilization.   From the artifacts and pictures, the explorers deduced that the   creatures who produced this civilization were four-legged beings with   a tentatcle that branched out at the end with a number of grasping   "fingers". After much study, the explorers were able to translate   Martian mathematics. They found the following equation:   $$5x^2 - 50x + 125 = 0$$   with the indicated solutions $x=5$ and $x=8$. The value $x=5$ seemed   legitimate enough, but $x=8$ required some explanation. Then the explorers    reflected on the way in which Earth's number system developed, and found   evidence that the Martian system had a similar history. How many fingers would    you say the Martians had?      $(a)\;10$      $(b)\;13$      $(c)\;40$      $(d)\;25$   P.S. This is not a home work. It's a question asked in an interview. Making sense of principal component analysis, eigenvectors & eigenvalues In today's pattern recognition class my professor talked about PCA, eigenvectors and eigenvalues.   I understood the mathematics of it. If I'm asked to find eigenvalues etc. I'll do it correctly like a machine. But I didn't understand it. I didn't get the purpose of it. I didn't get the feel of it.        I strongly believe in the following quote:     You do not really understand something unless you can explain it to your grandmother. -- Albert Einstein   Well, I can't explain these concepts to a layman or grandma.   Why PCA, eigenvectors &amp; eigenvalues? What was the need for these concepts? How would you explain these to a layman? Prove that if a function's average is nondecreasing then the function is nondecreasing. Let $\omega:\mathbb{R}_+\to\mathbb{R}_+$ such that $\omega(0) = 0$ and $\xi(s) = \frac{1}{s}\int\limits_{0}^s \omega(t)dt$ is nondecreasing, i.e., $\forall s,s': s\leq s'$, $\xi(s)\leq \xi(s')$. Show that $\omega(t)$ is nondecreasing a.e., i.e. find a function $w(t)$ such that $w$ is nondecreasing and $\omega = w$ a.e.  I was able to show the reverse implication easily, that if $\omega$ is nondecreasing then $\xi$ is nondecreasing. This way seems more difficult. How will people upgrade from 12.10 to later releases after 13.04 is EOL? If photons have no mass, how can they have momentum? As an explanation of why a large gravitational field (such as a black hole) can bend light, I have heard that light has momentum. This is given as a solution to the problem of only massive objects being affected by gravity. However, momentum is the product of mass and velocity, so, by this definition, massless photons cannot have momentum.  How can photons have momentum?   How is this momentum defined (equations)? Why are loops always compiled into "do...while" style (tail jump)? Prove that $f(x)\in \mathbb{Z}[x]$ such that $f(0)$ and $f(1)$ are odd has no integer roots How do I invoke a Java method when given the method name as a string? Stop execution of Javascript function (client side) or tweak it Using a foreign driver's license in California when staying with a J-1 visa I'm planning to attend a Californian university as a visiting scholar (not paid by the university) on a J-1 visa, for a total of four weeks; a colleague of mine will do the same, but for 12 weeks. I've been told (personally, ie. no citable sources) that, if you're staying on a J-1, you're effectively a California resident for the time you're there, and, therefore, need a Californian driver's license.  I've read up on this and found  that states:     If you are a visitor in California over 18 and have a valid driver   license from your home state or country, you may drive in this state   without getting a California driver license as long as your home state   license remains valid.      If you become a California resident, you must get a California driver   license within 10 days. Residency is established by voting in a   California election, paying resident tuition, filing for a homeowner's   property tax exemption, or any other privilege or benefit not   ordinarily extended to nonresidents.   I'm not planning to vote in any elections, nor will I own a home there, but I'm not sure what would constitute "privileges or benefits not ordinarily extended to nonresidents".  Do we need to get driver's licenses in California, or can we use our foreign licenses? How about day trips to Arizona or Nevada?  Note: this seems to be related to , but my question is specifically about staying there on a J-1. How prove this $\sum_{k=1}^{2^{n-1}}\sigma{(2^n-2k+1)}\sigma{(2k-1)}=8^{n-1}$ Given the positive integer numbers $n$,prove that $$\sum_{k=1}^{2^{n-1}}\sigma{(2^n-2k+1)}\sigma{(2k-1)}=8^{n-1}$$  where $\sigma(n)$ is defined as $$\sigma{(N)}=\sum_{d|N}d$$  This problem I think is very interesting, and have nice methods?  Thank you everyone.  and I find some of this     or can see: Any software for windows Writing custom (intelligent) microsoft windows firewall extension (C#/C++/C) Writing custom (intelligent) firewall for windows Installing nginx etc. on the same computer and use it as proxy with rules.   "You could not solve all Ddos problems with software" is not answer for this question.I want to reduce Ddos vulnarebility of my server.(It will stop low-level Ddos attacks)  Hosted solutions or cloud services are not related with this question.  Closing some of the ports and protocols via Microsot Windows Firewall is not related with this question.  Any idea?  EDIT After Questions  Low level: sample scenerio: 500 zombie clients and each of them sending 100 Custom Get/Post requests per second)   Web Application is on IIS. I want to stop attacks before they reach to IIS.  EDIT After Helpful Comment  How can the DDOS attack achieve its aim?   Exhausting bandwidth ( Not related with my question) Exhausting network hardware of server (NIC) ( Not related with my question) Exhausting server resources (RAM,CPU)  (Yep, I mean this)   If you stop attacks late, then you lost more resources! (Is this false?)   You can stop attacks on Web Application Level (in Asp.net MVc Code or in Handler) You can stop attacks on IIS (Dynamic Ip Restrictions)  You can stop attacks on lower levels (Firewall ..)   EXTRA INFO ABOUT THE LAST ATTACK (7 Hours ago)  When I connected server during the attack IIS Cpu Utilization was %92-%99. And when I try to connect homepage I got this error:  A connection was successfully established with the server, but then an error occurred during the pre-login handshake. (provider: SSL Provider, error: 0 - The wait operation timed out.)