Wikipedia:Reference desk/Archives/Mathematics/2006 November 15

= November 15 =

The phrase "quadratic in"
I am working through an example in a textbook and have reached the following equation:

$$y^2 - 2y = x^3 + 2x^2 + 2x + 3$$

The text then seeks to solve for y in terms of x, stating that this is a simple matter since the equation is "quadratic in y", and proceeds directly to:

$$y = 1 \pm \sqrt{x^3 + 2x^2 + 2x + 4}$$

I am unfamiliar with the phrase "quadratic in y" and do not understand how the solution was obtained. Any advice would be appreciated.

--202.168.12.132 04:04, 15 November 2006 (UTC)

I am not sure myself but maybe this happened
 * $$y^2 - 2y = x^3 + 2x^2 + 2x + 3$$
 * add one to both sides to "complete the square"
 * $$y^2 - 2y + 1 = x^3 + 2x^2 + 2x + 3 + 1$$
 * $$(y - 1)^2 = x^3 + 2x^2 + 2x + 4$$
 * $$(y - 1) = \pm \sqrt{x^3 + 2x^2 + 2x + 4}$$
 * $$y = 1 \pm \sqrt{x^3 + 2x^2 + 2x + 4}$$

152.3.72.50 04:20, 15 November 2006 (UTC)

"Quadratic in y" means "involving no powers of y greater than the second power y2". The origin of this terminology is related to the origin of the phrase "y squared", since "quadratus" is Latin for "square". McKay 04:55, 15 November 2006 (UTC)


 * Many thanks. My confusion is obliterated. --202.168.12.132 05:02, 15 November 2006 (UTC)

(after edit conflict) 'Quadratic in y' means that if you treat every other symbol but y as a constant, you get a quadratic equation. In other words, all terms involving y are either (something).y or (something).y^2, and not (for example), y^5, 1/y, ln(y), sin(y) or e^y.

The solution given is the normal quadratic solution $$ (-b \pm \sqrt{b^2 - 4ac}) / 2a$$ with substitutions: --ColinFine 05:06, 15 November 2006 (UTC)
 * $$a = 1$$
 * $$b = -2$$
 * $$c = -(x^3 + 2x^2 + 2x + 3)$$

presenting our firm's finacial health before vanture capitalist
how to present financial health of the company in front of vanture capitalist to understanding them to invest in the company? what are the different tools to be used?


 * I don't think they would be as much concerned with the current financial health of the company as much as it's potential for growth. After all, even if you are in bankruptcy, they could fix that, if they see themselves making a nice profit by helping you out.  You should, however, have a detailed budget of what you intend to do with the money they invest.  StuRat 07:06, 15 November 2006 (UTC)


 * This is not really a mathematics question. Try Reference desk/Miscellaneous. Think from their point of view: Why would they want to invest in your company? Basically you have to show that the expected profit is (much) larger than the money to be invested. What is your elevator pitch? One advice: if you are presenting in English, have your presentation checked for correct spelling and grammar. --Lambiam Talk  07:21, 15 November 2006 (UTC)


 * You will need to be able to explain clearly:
 * How much you want the VC to invest
 * What you will use the money for
 * How much equity you are prepared to give the VC in exchange for their investment - be prepared to negotiate on this
 * How long you will need this investment for - VCs are typically looking to invest for between 2 and 10 years
 * How much you think the company will be worth at the end of this period - this is the hardest part, as you will need to show credible financial projections and demonstrate that you understand your market, your competition etc. The VC will try to talk down your valuation - this is a negotiating tactic to make a case for receiving more equity - so you need to have confidence in your projections. Gandalf61 08:32, 15 November 2006 (UTC)


 * The venture capitalist would almost certainly expect to see a business plan. I expect there are articles on financial ratios and discounted cash flow you should look at. If I was the VC, I would be mostly interested in what my rate of return on investment would be - so it would be good to be able to give a projected internal rate of return. In the UK, and probably in North America and other countries, there are government funded bodies that can give you advice about this. You could talk with your accountant about this, but they will probably charge you a lot of money. There are many books published about writing business plans, plus a lesser number about pitching to venture capitalists.
 * I've just had a look at the financial ratios article, and it could be improved. But any textbook on management accounting will cover financial ratios. If you can, I suggest a trip to a public library to see what books they have. Do not be afraid to tell the librarian what you want to do and ask their help.

Tetromino Game
Suppose there is a game where you have an 8x8 grid that can only be filled with tetrominoes and the game ends once no more tetrominoes can be placed on the grid. What is the maximum number of squares that could be unfilled by the end of a game? --Tuvwxyz (T) (C) 22:23, 15 November 2006 (UTC)


 * After a bit of playing I've got 20. But I wouldn't know how to prove this mathematically. ;) Vespine 00:21, 16 November 2006 (UTC)


 * Hmm, just as I thought I was getting a theory I jsut came up with 28... Vespine 00:33, 16 November 2006 (UTC)


 * I managed 24. I'm not sure much better could be done than 28, but I can't think of any way to prove any of this other than an exhaustive search. Wasn't Tetris proven to be NP-Hard not so long ago? - Rainwarrior 02:34, 16 November 2006 (UTC)

An equivalent formulation would be: What is the minimum number of tetrominoes required to fill the board such that no more tetrominoes can be placed?

This may have a nice graph theoretic equivalent as well - I'm not too sure what it would be, perhaps a matching problem on a graph which represents the grid, and finding a minimal subgraph with certain properties? I'd have to think this one through a little bit more...--ManicLogic 02:33, 16 November 2006 (UTC)


 * I still think my 28 solution is correct. SPOILER: The 8x8 is split up into four 4x4 groups which are symmetrical about the centre. The very middle is occupied by a 2x2 square block, with each 4x4 corner quadrant containing only two T sections arranged back to back and offset by one, such that the 3 units adjacent to the corner are vacant. I hope that's a clear enough description to recreate it. Vespine 03:54, 16 November 2006 (UTC)


 * Is this what you mean?

. . t. . T. . . T t t T T t. T T t. . T t t . T. O O. t. . t. O O. T. t t T. . t T T . t T T t t T. . . T. . t. .
 * }
 * Here is 'my' solution:

. I I I I. . . . N N. . L L L N N. J. L. . T. . J I. O O T T J J I. O O T. . . I. T. . T T T I T T. . . T. . . T.
 * }
 * It's not as regular as yours, but gives the same result: 28. --CiaPan 16:59, 16 November 2006 (UTC)


 * Yes, that's what I came up with. Nice ASCII. :) Definitely still have no idea how to try to prove it! I started to think that 1/3 was the most that could be empty since the pieces of 2x3 area still fill only 2x2 area, leaving 2 units of area empty, but that's obviously wrong, it doesn't take into account leaving empty shapes into which tetrominos can’t fit, like the L shapes made of 3 units I've left in the corners, and the several 1x3 units you have in your pattern. Vespine 21:25, 16 November 2006 (UTC)


 * A thought... at most you can only have 3 connected empty spaces, and it would take at minimum 8 blocks to surround any 3 connected spaces. However, any boundary of blocks may be shared by two empty spaces, so perhaps this can be done with only four blocks per space. Thus, approximately 3/(3+4) of the space may be filled. With 64 blocks: 64*3/7 = 27. Additionally, the constraints put on the shapes of the tetrominoes (having to be vertically or horizontally connected) probably lowers this number (call it "estimated density of a maximally open solution" or something) a bit more, but the fact that the board has a closed edge counteracts it in the opposite way. This is light-years away from a proof, but maybe a reasonable explanation? - Rainwarrior 23:12, 16 November 2006 (UTC)


 * Maybe if I attacked the problem of how many open areas a square can touch. Either it's 0 (completely surrounded), 1 (a corner), 2 (a line), or 3 (an end). Since there are no tetrominoes of a single clock, there is no 4. It seems though, that you cannot introduce a 3 without indroducing a corresponding 1 elsewhere, which means that 2 is the best you can do on average, supporting my earlier idea. Hmm... Anyhow, I finally found my own 28:

. S S. . L. . . . S S. L. S L L L. L L S S .. L O O. S. . S. O O L. . S S L L. L L L S. L. S S. . . . L. . S S.
 * }
 * (You can use an L or anything else really to fill the middle hole instead of the O). - Rainwarrior 01:01, 17 November 2006 (UTC)

I wrote a computer program to take a look at this, but I think it would take about a month for this computer to enumerate all of the cases (searching every configuration of 9 or less pieces, since we know the best solution to be not worse than 9 pieces). So... interestingly... an answer by exhaustion is possible. I just can't give it to you at the moment. It will take time. Ha ha ha. - Rainwarrior 02:30, 17 November 2006 (UTC)


 * Oh wait, I missed one of the factors in that calculation. Change "month" to "forever". Nevermind. - Rainwarrior 02:41, 17 November 2006 (UTC)