Quadratics

[Tonic Please]

2X (squared ) - 7X - 15X = 0

The possiblities for X are:


3/4 (as in 3 OVER 4)

and

1.5

What is the solution? I have been on it for 3 hours and it/s bloody annoying.

Many many thanks in advanced.

Smooth skies,

Dan. :D

[jorgvaz]

Ok...Dan, try this,


2x2 –7x –15x = 0

2x2 = 7x + 15x

2x2 = 22x

x2 = 22x
2

x2 = 11x

11*11= 121
x = 121

2(121) – 7(11) –15 (11) = 0

[john_tullamarine]

Perhaps you could revisit the quadratic roots solution ...

for f(x) = ax^2 + bx + c

the roots are

x1 = (- b + (b^2 - 4ac)^0.5)/2a

x2 = (- b - (b^2 - 4ac)^0.5)/2a


I presume that the 15x is a typo and you meant to write

2x^2 - 7x - 15 = 0

which gives roots (assuming that I have counted all my brackets correctly and put in the correct numbers ...)

x = (-(-7) +/- ((-7)^2 - 4(2)(-15))^0.5)/2(2)

= (7 +/- 13)/4

= 20/4 and -6/4

= 5 and -1.5


If what you wrote is correct and what was intended then the equation is a quadratic whose constant term is zero and the solution comes down to zero and 11.

[Tonic Please]

Correct. 15 not 15X.


And Jorgvaz....your answers seem different to the ones in the book. Is that because of my 15X and not 15 alone?

It is 2 "ex" squared minus 7 ex minus fifteen, so you can see how I mean. The X may mean times to some but it is X not times...

Thanks so far people. Appreciate your efforts! Just not exactly the answer in the book, possibly my mistake.

Dan :)

and John, everything you said is correct about the root values....sorry, I should have re-written my questrion..thanks for understanding anyhow...

[john_tullamarine]

Maybe the explanation of "completing the squares" to find the general quadratic solution was not given to you clearly.

One of the math references (http://mathforum.org/library/drmath/view/60700.html) in the Tech Forum sticky thread gives you the background ....

[Tonic Please]

Cannot thank you enough. I can look at any problem and understand it, but when it comes to trying to do it for myself with different numbers, or a slightly different "appearence", I just lose myself.

Many thanks

Dan :cool:

[john_tullamarine]

Dan,

No different to learning to fly .... mainly a matter of confidence and practice ...

(a) demonstration .. the teacher (instructor) does the sum on the board (demonstrates a landing)

(b) practice .... you try the same problem and then with different numbers under the supervision of the teacher (instructor might follow through on the controls and then you do it yourself with the instructor monitoring)

(c) independent practice .... you do some sums on your own (solo circuit practice)

(d) competence ... you can do any similar problem with confidence (you get your PPL)

Stick with it, buddy .. just a matter of thinking about it, practising it, gaining confidence and competence in doing it.

[kabz]

Just as a wild shot, draw the curve (for that's what it is) out on some graph paper, or a computer program. Should be some Excel stuff for this ...

Then if I remember right, the roots are where the curve crosses the x-axis.

I'm soooooooo happy to have left 2nd year uni maths behind... :mad:

[Genghis the Engineer]

The art of understanding (rather than simply solving) quadratics is to factorise them, thus...

2X²-7X-15 = 0

(divide through by 2)

X²-3.5X - 7.5 = 0

This must have a form (X+a)(X+b)=X²+(a+b)X+ab=0

Or in other words, a*b=-7.5, and a+b=-3.5

Play around with numbers a bit, and you'll see that this works for a=-5 and b=1.5, ziv...


(X+1.5)(X-5) = X²-3.5X-7.5 = 0

[or if you prefer, 2(X+1.5)(X-5)=2X²-7X-15]


And since (X+1.5)(X-5)=0, there are only two answers, since one of the brackets must equal 0.

X=-1.5, X=5.


QED.

G

Kabz, is this 2nd year degree level in the US? At my English Grammar school I was doing this when I was 14. The second year of my degree was more along the lines of complex calculus in multiple variables. Now that I am glad not to have to do any more, 2nd order differential equations is about as hard as it gets in the real world.

[Tonic Please]

thanks everyone, for the solutions and links. It is very helpful...I guess you have to start learning somewhere, until you get the momentum going.

Thanks again :D :D

Dan :)

[ATPMBA]

Quadratic Equation

2x² – 7x –15

I use the FOIL method, terms of the equation are multiplied together and go in the following order.

F)irst
O)uter
I)nner
L)ast

To factor the equation set it up in the format below. The numbers and signs +- need to be tweaked. The terms are multiplied together, the Outer and Inner are multiplied and then added, Last is multiplied. For the First we know x times x equals x² and we need a coefficient of 2, so put in 2x. And we need some numbers that when multiplied equal –15. A little bit of trial and error is needed.

Format
( x + _ ) ( x - _ )

First 2x x
Outer 2x -5
Inner +3 x
Last +3 -5

Factored (2x + 3) (x - 5)

Check the results:

First 2x²
Outer -10x
Inner 3x
Last -15

Showing all the work as not to lose points.

2x² - 10x + 3x -15

Simplified:

2x² -7x – 15

To answer the question of (2x +3) (x – 5), the answers are –1.5, 5.

Dedicated to Mr. Barrett, my algebra teacher.

[Blacksheep]

Hey Ghengis you must be younger than I thought! When I was at Grammar School, Northern Universities Joint Matriculation Board Syllabus 'B' introduced us to differential calculus in the fourth year. A great boost when we moved on to integration of multiple trigonometrical variables on the Ordinary National course when I was an electrical apprentice. I always knew that standards were dropping, but differential equations definitely aren't in any university syllabus that I'm familiar with either. ;)

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Through difficulties to the cinema

[Genghis the Engineer]

Maybe you're older than you thought Blacksheep? :D

It was some time ago, but I'm reasonably sure I did quadratics and simultaneous equations in the 3rd year, a very basic mention of differentiation in the 4th year before doing O-level, then rather more calculus in the 5th year for AO level. A level got as far as Laplace transforms, second order differential equations complex algebra and quite a lot of mechanics and stats. First year degree was mostly more of the same but deeper with more applications and lots of numerical modelling (starting at Newton-Raphson and progressing upwards), and second year degree a lot of fairly obscure methods such as the aforementioned calculus of complex numbers - which I'm afraid rather passed me by although I suppose I must have passed the exam at some point.

I'm not that young, I did go to a Grammar school before the worst excessives of certain lefty governments decided to turn them all into the comprehensives that my kids have to suffer because I can't afford to send them elsewhere.

G

[Blacksheep]

Ha-haa! Ghengis, my knees and back keep telling me that too! :D

The brain cells don't hurt much yet, though. I haven't used the calculus of complex numbers for a long time. It seemed very important at one time, but I can't remember the last time I ever used it in anger.

Did I say "can't remember"? Oh Dear, perhaps the brain cells are older than I thought eh? ;)

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Through difficulties to the cinema

[Dimensional]

Okay, the classic accusation that A-Levels are getting easier ... :mad:

I'm doing A-Level maths now, and we do first and second order differential equations, polar and parametric co-ordinate systems, hyperbolics, matrices, complex calculus (in the Argand plane), and numerical methods (Newton-Raphson) as well as plenty of mechanics (SHM, circular motion) and stats, we did differential calculus in the fourth year for GCSE coursework, integration in the fifth year ... admittedly we are doing further maths, but hey :D

(and yes, I might well be going to Impossible (sorry, Imperial) College to do Aero Eng, and yes I know it's mainly useless in the "real world"....:cool: )

So stop accusing A-levels of being easy (I hasten to say "easier"), because they aren't!

-D

[Genghis the Engineer]

Calm down old boy, so far as I could see the only pokes were at Americans (always fair) and Comprehensives (often fair).

What you are doing sounds very similar to my further maths A level some time ago. I imagine you are doing more Newton Raphson, etc. than I did because that's a big player in computer analysis, and some stuff has been dropped off to balance the syllabus.

Now Imperial, well I could start on them, but probably only on principle because I'm a Southampton graduate....

G

[Dimensional]

Sorry :) A little quick to anger, since I was just a tad fed up with someone denigrating A-Levels before I came on here ...

And in fact, it looks like I'm not going to Imperial, but Cambridge, despite the fact that their course is rather dire, it *is* Cambridge and I don't fancy living in London for four years... (I do really like the Imperial course, maybe I'm a masochist or something...)

-D

[Flash2001]

They didn't ask you to solve a quadratic, only to establish which answer fits. Try the answers provided by substitution! JT's work of course.

After an excellent landing you can use the airplane again!

[Genghis the Engineer]

Many years ago I had an interview at Cambridge for Aero-Eng, it became very clear halfway through that they didn't want to teach what I wanted to learn, and I didn't want to learn what they wanted to teach.

So I went to Southampton, and learned a great deal. But without a doubt, Cambridge is a much prettier city all else being equal.

G

[Dimensional]

I'm quite lucky -- the interviews at my chosen college (Trinity) were done by a fluid dynamics and a mathematics proffesor :cool: And also, I wouldn't mind doing all the other engineering stuff *and* still have the chance to specialise in Aero Eng (aerodynamics / CFD etc).

I'll admit to having fallen in love with the place. And RAF Wyton (home of ULAS and CUAS) is a damn sight closer to Cambridge than London ...

[VnV2178B]

Hold a bit Dimensional,

Genghis and Blacksheep are reliving their own schooldays - roughly the same as mine from the topics discussed - no-one mentioned the relative difficulty of today's A levels compared with what we put up with. Your list seems to similar to the NUJMB 1968 sylabus as far as I recall so no change in difficulty there then!

Best of luck with the further maths - I tried and failed !

VnV...

[Notso Fantastic]

By Jove I enjoyed that! Dusted off the old 'O' level maths a treat, and the answer popped out like a sweetie. The secret with quadratics is like 'how do you get to Carnegie Hall?' Practice practice practice. Do them time and again, and again......and again! But don't expect that to work for calculus- might as well be in Chinese!

[VnV2178B]

Genghis,

you could be me !

I considered Cambridge and ended up at Southampton too, mind you I went for electronics not aero...

All this talk of maths just serves to remind me that I can't remember using any of it since! But I suppose it is valuable to know, just in case someone crops up on PPRUNE and asks.

VnV

[Genghis the Engineer]

To be fair, although Cambridge and I considered each other and came to similar conclusions - they did get in the first strike.

I do use maths, mostly algebra, simultaneous equations and numerical methods a lot. Calculus fairly regularly, differential equations about annually (guaranteeing that I always have to get the books out for those). Complex numbers, not since ETPS.

G

[Captain Stable]

I started off in Nuclear Physics in a Uni that was not so much red-brick as pre-stressed concrete. The nerds who really got on my nerves were the computing & cybernetics guys who used to do calculations in octal or hexadecimal... :rolleyes:

[john_tullamarine]

What ? .... octal and hex ? .... doesn't anyone do things proper these days and use binary ?

[Genghis the Engineer]

The world is divided into 10 types of people; those who understand binary and those who don't.

(Actually not original, I nicked the line from Jeremy Paxman, quite incisive on his part I thought given he's only got an English degree.)

G

[john_tullamarine]

G,

Love it ......

Having started out on mainframes, I then started playing with the earliest crop of micros in the mid-70s (was it really that long ago ?) .. the only way to get anything done was to work low level ... almost all of it, of course, is now just a very, very vague memory. Now that PC and software houses have gone the way of an every year upgrade to force people to scrap the old gear and software which did a perfectly good job (albeit with not so many unnecessary, but flashy, bells and whistles), I guess that not many people at the user level bother with low level programming any more .... ?

I vaguely remember, as an undergrad, the department getting a PDP-whatever, which had, if I recall, 8k of RAM, a paper reader for input, and a teletype for output. Not to mention an HP-whatever, which had something in the order of a few hundred memory steps. ... now I have to pull out the manual to work out how to use the old slide rule ... sad, isn't it .... dinosaurs forever !!

[Iron City]

Twas probably a PDP-8. I graduated before our high school got one so had to learn programming on a time share system using teletypes and 8bit paper tapes as a storage medium.

The university curricula sound about the same as American undergraduate programs, and I believe that includes Texas except maybe Aggies.

[Genghis the Engineer]

For the benefit of foreign readers, A levels are taken at age 17 or 18 and are the English University entrance exams. (O levels were taken at 15 or 16 and are now called GCSEs).

G

[Onan the Clumsy]

I vaguely remember, as an undergrad, the department getting a PDP-whatever, which had, if I recall, 8k of RAM, a paper reader for input, and a teletype for output.

You were lucky! We used to dream of that.

Actually, that's exactly the machine I first learned on. A PDP-8e that had been upgraded from 4k to 8k with a paper tape reader and a teletype.

I made a tape once that when you ran it through the reader and echoed it to the teletype, it printed out the football results. I embeded a lot of NULLs in there too, so the print head bobbed up and down a lot and it looked just like the teletype on Grandstand.

It was a sign of things to come.


Oh yeah and not that A levels are so much easier now or anything, but this was back in junior school :)

[john_tullamarine]

... ahhhh ........ dinosaur memories .... I think that you might be right, chaps ... PDP-8 rings a vague bell in the rapidly decaying grey matter .... as I recall, in our spare time, we all developed high skill levels in causing the paper output to print pictures of dubious qualities ....

[foghorn]

The way we're going we're going to be solving the Navier Stokes equation next...

Dimensional - Trinity is a good college, tends to be a bit insular because of its size, it's just a shame that 1st and 3rd (Trinity Boat Club) is crap.

Which fluid dynamics lecturer interviewed you - if they've been around for a bit they probably had a displeasure of teaching me?

cheers!
foggy (Selwyn College 1991)

[Captain Stable]

Uh oh. :rolleyes: I may be the baby of the family here. In my day (this was at university) we had a couple of PDP11's interfaced with an IBM4130. Still had teletypes. I remember the first time I saw a dot matrix printer. I thought "SH!T that's fast!". <sigh> Lost innocence!

[Blacksheep]

Did someone mention dinosaurs?

I still have my original Faber-Castell ENG-II portable. It was lightning fast that old "Ziggy stick" although the processor speed wasn't matched by a decent RAM capacity. It could only store the current operation and there was no cache. This wasn't too much of a disadvantage - you could always do a hard copy data dump at regular intervals. At least it worked anywhere, anytime and had no batteries or power supply to go wrong or die on you at a critical moment. Finally it could be used to draw a decent straight line or even measure things, as long as you were prepared to count in 32nds of an inch - none of that decimal nonsense on the British model of the good old Faber! I can still solve complex trigonometrical functions on it (power calculations generally) faster than most people can type the input data on a keyboard.

Its hard to imagine that until the late seventies, almost all design calculations were done with a slide rule. :D

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Through difficulties to the cinema

[VnV2178B]

Blacksheep,

WOW a Faber-Castell!!!, I used to yearn to have one of those, ended up with a British Thornton , good enough for most things up to A level but the plastic cases were naff (I still have it, together with a neat wooden box that my dear old dad made for it).

I also remember going to a maths course for a week at Nottingham University while I was doing As where we used mechanical 'wind-up' calculators to do the stats. section.

Computer-wise, cut my teeth on an Elliot 803 then progressed to a PDP-8i: a six-foot rack of it. When the later PDP-8e came along we were amazed when a bloke turned up with a box under his arm and said 'where do you want your new computer putting,
then?'

But slide rules and computers can't really DO quadratics can they?


VnV...

[Genghis the Engineer]

My 12" British Thornton still lives on my desk and gets used for occasional light relief or when somebody's borrowed my calculator.

As it happens, yes, the higher level Casios will solve quadratics and simultaneous equations - a facility that I probably use at least annually ;-) When I did some AFCS design work once I found the high end casio calculator's ability to manipulate 6x6 matirices particularly useful. With the best will in the world, and much as I love it, my slide rule just doesn't do that.

G

[Blacksheep]

Going back to the mathematics - boring old quadratics and stuff - I find that statistics is far more important in everyday Technical Services work than any of the stuff I learned back in basics. Given the importance of statistics in reliability analysis, I often wonder why we were taught only the most rudimentary statistics in our engineering training - a PhD student of Linguistics knows more about statistics than the typical engineer. My first year economics started with a book called "Elementary Statistics for the Social Sciences" The first eight chapters were fairly easy reading, the remaining twelve took me to places I'd never been before. Then came Advanced Statistics! I find that I use all that stuff more than any of the mathematics that I learned in engineering studies.

I'd like to see more statistics included into the secondary school syllabus - its used in almost every field of endeavour today but is second only to sociology in the list of the most misunderstood subjects.

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Through difficulties to the cinema

[Dimensional]

Ackkk .. anything but statistics! :yuk:

[Anthony Carn]

Statistics ? ---- "A numerical form of wish fulfillment". Just joking ! (an average joke ?).

I have a cylindrical slide rule, found in a junk shop. No identification, except "N 1010" on the bottom end, "OTIS KING'S PATENT No 183723" on the scale and "Made in England" on the top end. Steel hand hold, brass sleeve, ivory plastic scales. 6" compressed, 10" extended, 1" diameter.

Does anyone know of the whereabouts of a set of instructions, please, either for this model, or just a general description e.g. on the net ? :confused:

Apologies is off-topic, but you already seem to have got to the root of the quadratics question.

[Genghis the Engineer]

I'm guessing you have something like http://cgi.ebay.co.uk/ws/eBayISAPI.dll?ViewItem&item=2504046369&category=361 which is quite a rare collectors piece.


Try the following...

http://www.mat.bham.ac.uk/C.J.Sangwin/Sliderules/sliderules.html

http://www.sliderule.ca/


I have to confess, I have a small private collection of interesting slide rules, but an Otis King cylindrical has very sadly somehow escaped me. Amongst aviation slide rules I own are...

USAF type CPU-73A/P "Attack computer"

RAF Lifeboat site computer Mk.1

A small 6" log-log rule stamped "Dowty aircraft components, Cheltenham"

A 6" specialist slide rule for calculating air flow rates, stamped "Lucas Gas Turbine Equipment"

A Scout Helicopter "Flight Envelope computer".


There, my secret's out, I am a very sad individual with very odd hobbies.

G

[Anthony Carn]

Genghis the Engineer

Many thanks for the information -- you're a star !

It is indeed the item you suggest, with the exception that the "sleeve" is brass on mine, as opposed to black (plastic?) on the one you refer to.

Otis King manual duly downloaded from www.sliderule.ca/manual.htm ! YEY ! :D :cool:


PS -- Don't think you're sad at all ! Some of these slide rules are very lovely items ! :)