the tree

It turns out this is actually a thing, and you can sort of get something from it. Still no name for it though.

I think it covers all the conic sections, although I really can't see any merit to writing them like that.

It doesn't really represent anything. Are those supposed to be multiplied together? Are we talking dot multiplication? Even that only makes sense if there are the same amount of rows on the left as columns on the right of the dot. edit Okay, it occurs to me that you might have meant [math]\left[ \begin{array}{ccc} x&y&1 \end{array} \right] \cdot \left[ \begin{array}{ccc} a&b&c\\d&e&f\\k&m&n \end{array} \right] \cdot \left[ \begin{array}{c} x\\y\\1 \end{array} \right] [/math] Which does indeed expand to the expression in the OP. In which case, no name, just a very obscure and typo ridden way of writing the general form of a two variable polynomial of degree two. It doesn't represent an equation. It's an expression. Equations have equals in them.

Part of the trick with modelling is to know what sort of answer you're looking for. Exact answers like "at time T, the value x will be f(T)" require an entirely different approach to holistic answers like "the system oscillates under these conditions, and doesn't under these", neither approach is better or worse since you can't expect outright accuracy from any model. But yeah I have no idea what we're being asked either.

Okay you're not going to progress with out either learning polynomial long division or using a calculator. Now just to move on, we might just have to tell you. [math]\frac{4 x^3+23 x^2+34 x-10}{x-(-3+i)}=4 x^2+(11+4 i) x-(3+i)[/math] To find the two remaining roots, you need to solve that quadratic equation. If that is posing a major difficulty then you are trying to learn too fast and you need to go back and learn how to do that. In the particular case, the root you gave is indeed one of the remaining roots. There is one more needed to make a total of three.

To make this as clear as conceivably possible: an nth degree polynomial has n roots. You are dealing with a 3rd degree polynomial, you have one root, you need 2 more. So you need to solve a 2nd degree polynomial, you have hopefully known how to do that for a very long time.

In other news, blue things are not red.

If you're looking a greater degree polynomial than you started with then you are definitely going wrong. The two things you need to get right in this question are polynomial long division (for which there will be no remainder) and then solving a quadratic equation. That's it. Don't try to over complicate that.

There are three matrices there. Three. THREE.

I don't think the dotted lines are definitely asymptotes - if they were then a and b wouldn't be representing anything.

This does happen, text books are fallible. But that is a pretty awful failing, I'd concur with DrRocket.

I suppose that on a poincaire half plane you can essentially only go left, right or up. Is that what we're talking about?

Although it is very, very, very important to know that the two definitions are equivalent. @Juilingstar177, I would strongly recommend that if you are interested in logic then you begin to study it properly. It is a rich, beautiful, complex and powerful field of study. But you will not be able to study it just by making assumptions and running with them, actually you wont be able to study anything that way. For starters, you trip up fairly early on, trying to prove a definition (which is somewhat like trying to scribble over a bucket of black paint with a black pen in order to make it black). Actually, no, there's nothing to be proved - it's just part of the definition of "=". It's called reflexivity and alongside symmetry and transitivity we can define an equivalence relation which "=" is the canonical example of and yes a pretty damn important tool in mathematics but a pretty uninteresting one on the whole. I think this has been fairly well covered. Just don't mix up physics and mathematics - in short mathematics is logical and the universe is well, not. I would strongly advise you to simply research the definition of division in the context of mathematics, you'll be kicking yourself fairly quickly. And try not to make analogies about apples - only one guy ever gained any credibility doing that and he's generally considered a bit a dick nowadays anyway. Actually far from it, formal logic has had a much more significant role in mathematics in the past century than it has ever had since the days of Plato, I'm really surprised people aren't more aware of this - just think what all the software on your computer is made out of! If you really want start appreciating the places that logic and mathematics can take you - then start asking questions, reading, thinking, asking more questions and then challenging the answers you receive. You can of course declare simply how you think things are and feel smart because you used the word "logic" (which, for the record, a lot of us hear every day). But if you take a moment to learn how things actually are then prepare to have your mind blown, because they are so much bigger and so much more complex than any of us. You will never, ever want to go back to living in the dark.

It's not always entirely a co-incidence, and perhaps probability theory isn't what you need to explain this. As a six year old, there's a high chance that your daughter is already odly aware of the Baader-Meinhof Phenomenon, even more so than you are. When you're learning new words every single day then it is often something that you really become aware of. The thing to consider here is, if she hadn't recently been thinking about dalmations - she may well not have noticed the dalmation in the next car. Seeing a dog in a car is a completely unextraordinary event, but the reason for this story happening is that she was probably subconsiously on the look out for black and white spots.

Yeah that definitely intersects (see graph), but at [imath]p=10[/imath], (graph) the line is clearly tangent to the curve. Where you majorly went wrong: Check your knowledge of solving inequalities. As welll as the inequality being the wrong way round, this is wrong in another way. Which you should be able to figure out by, y'know, looking at the graph.

I'm assuming you meant separation of variables, and then integrating twice, right?

Actually the general sollution is the sum of the homogeneous and implicit solutions. Rearranging it like so will make it a little easier: [imath]x'' +1 = e^x[/imath] You should be able to solve the homogeneous equation [imath]x'' + 1=0[/imath] without much difficulty. Finding an implicit solution is essentially just well informed guesswork - what do you think the solution will look like?

Yeah, I think it's quite a common mistake as most people have been using [imath]\sqrt{ab}=\sqrt{a}\sqrt{b}[/imath] for long a time before they are introduced to complex numbers.

Guys, the problem is a lot simpler than you're making it out to be. This step: [imath]\sqrt{(-1)(x-1)}=\sqrt{(-1)}\sqrt{(x-1)}[/imath] just isn't right. In general [imath]\sqrt{ab}=\sqrt{a}\sqrt{b}[/imath] is only for positive reals: try [imath]a=b=-1[/imath] for instance.

If there is a proof not reliant on Cayley Hamilton, then I can't see it. I imagine it would be acceptable to take that theorem as a given for the sake of homework.

Writing at the knows-some-math level is particularly difficult, you don't want to get bogged down in technicalities but your readers may get irritated if you aren't concise and try to use analogies in place of precision. I'd say any amount of new terms and new notation (new to the reader, obviously) is completely fine on the conditions that: You explain each new thing. You introduce them one at a time. No-one likes reading more than a couple of definitions in a row and then trying to comprehend them all for the first time in the next example.

The simplest, crudest approach is to write it as a set of linear equations and look for a solution from there. If [imath]c_1 A_1 + c_2 A_2 + c_2 A_2 = A[/imath], then that implies four equations that can be reduced using good old fashioned elimination. The more mathsy approach would be to summarise the the set described by [imath]\mbox{span}( A_1 , A_2 , A_3 )[/imath] in such a way that it's obvious whether or not [imath]A[/imath] is included. As Acidhoony said, that set is going to be at most 3-dimensional.

The first question is obvious. What makes you think that anything in your original post is correct? Beyond that, any questions I would have about gravity would ask for predictions that ideally would be testable or at the very least not contradict extremely well known evidence.

I think it's fair to say that Einstein is already spinning in his grave from that. So what are these things for? Or, for that matter, these? Well we'd certainly notice if we were dramatically wrong about that. Planetary orbits wouldn't be the right shape for a start. 'g' in terms of classical mechanics usually refers to the rate of acceleration due to gravity at the Earth's surface, rather than any force itself. Considering that Newton's description of gravity predicts both the behavior of objects close to Earth and the heavenly bodies, and that such a thing isn't particularly difficult to calculate, what possible reason is there to presume that two different things are acting? Why would we only get a small portion of the Earth's gravitational oompf when say, one of us in low earth orbit gets the whole lot? This is a good example of where you should think for yourself, before typing something like that.

The correct answer is pretty definitely x=1. How did you get x=2?