Quadratics
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Quadratics
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.
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.
Moderator
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.
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.
Last edited by john_tullamarine; 6th Jan 2003 at 22:42.
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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...
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...
Moderator
Maybe the explanation of "completing the squares" to find the general quadratic solution was not given to you clearly.
One of the math references in the Tech Forum sticky thread gives you the background ....
One of the math references in the Tech Forum sticky thread gives you the background ....
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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
Many thanks
Dan
Moderator
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.
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.
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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...
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...
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.
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.
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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
Dan
Thanks again
Dan
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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.
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.
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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. 
**************************
Through difficulties to the cinema
**************************
Through difficulties to the cinema
Maybe you're older than you thought Blacksheep?
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
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
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Ha-haa! Ghengis, my knees and back keep telling me that too!
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?
**************************
Through difficulties to the cinema
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?
**************************
Through difficulties to the cinema
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Okay, the classic accusation that A-Levels are getting easier ...
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
(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".... )
So stop accusing A-levels of being easy (I hasten to say "easier"), because they aren't!
-D
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
(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"....
)So stop accusing A-levels of being easy (I hasten to say "easier"), because they aren't!
-D
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
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
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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
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
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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!
After an excellent landing you can use the airplane again!
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
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
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I'm quite lucky -- the interviews at my chosen college (Trinity) were done by a fluid dynamics and a mathematics proffesor 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 ...
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 ...