[gkathotia]

Hi JUNTA
Me back from the basics just like the PHEONIX . And guess what ?
I have jotted down some 300 odd important /subtle formulae concepts in QUANT which could bolster accuracy and speed(sounds like CAT elixir , eh) . It will be my priority to share these concepts and important formulae/concepts via this thread for you all .
Suggestions,errata,additions are most welcome (along with some praise :wink: )
Let's hit it then !

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04/10/2002
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THEORY OF EQUATIONS:
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(1) If an equation (i:e f(x)=0 ) contains all positive co-efficients of any powers of x , it has no positive roots then.
eg: x^4+3x^2+2x+6=0 has no positive roots .

(2) For an equation , if all the even powers of x have some sign coefficients and all the odd powers of x have the opposite sign coefficients , then it has no negative roots .

(3)Summarising DESCARTES RULE OF SIGNS:

For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x) .
Hence the remaining are the minimum number of imaginary roots of the equation(Since we also know that the index of the maximum power of x is the number of roots of an equation.)


(4) Complex roots occur in pairs, hence if one of the roots of an equation is 2+3i , another has to be 2-3i and if there are three possible roots of the equation , we can conclude that the last root is real . This real roots could be found out by finding the sum of the roots of the equation and subtracting (2+3i)+(2-3i)=4 from that sum. (More about finding sum and products of roots next time )


WELL , THAT'S ALL FOR TODAY. Will be back on monday with more crucial concepts. Meanwhile lets all screw SIMCAT 3 !
Chiaaao
Gaurav

edited by pagalguy : converted to a sticky post

[archierao]

hey thanx for sharing these. i bet all will find it useful while revising quant topics.

cya!
archierao

[pagalguy]

Too Good Gaurav. Thanks for taking the time to educate us all over here ! I'm sure each and every soul appreciates what you have done here and planning to do

Warmest Regards,

[capreal26]

cooooooooool ones Gaurav.......

great work... thanx a quintal (since tonnes are very heavy. somebody said on this forum :P ) ........

eagerly waiting for your next edition.......

- capreal

[Emperor]

Very good ones.

Will definately help

[ravi]

Thx a million, Gaurav !I had been 2 lazy 2 go back and revise those awful formulae. Thx 4 takingthe trouble 2 put it up. Btw, i didn't get any feedback from any of the regulars (u,pagal etc ) on my DI shortcut 4 calculating %s. Did u miss the thread?
ravi

[pagalguy]

Gaurav,

Update this thread often & I'll put this one up as a sticky. This is one super thread and thank you so much for the effort you are putting into this one. My heartfelt thanks. You are truly helping all of us out at the same time.

Regards,

[gkathotia]

Thanks for all the unworthy adulation( :wink: ) . Actually there is my personal gain from this section. It helps me revise the formulae and concepts. '
Anyway , today's concepts:

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07/10/2002 THEORY OF EQUATIONS
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(1) For a cubic equation ax^3+bx^2+cx+d=o

sum of the roots = - b/a
sum of the product of the roots taken two at a time = c/a
product of the roots = -d/a

(2) For a biquadratic equation ax^4+bx^3+cx^2+dx+e = 0

sum of the roots = - b/a
sum of the product of the roots taken three at a time = c/a
sum of the product of the roots taken two at a time = -d/a
product of the roots = e/a

(3) If an equation f(x)= 0 has only odd powers of x and all these have the same sign coefficients or if f(x) = 0 has only odd powers of x and all these have the same sign
coefficients then the equation has no real roots in each case(except for x=0 in the second case.

(4) Besides Complex roots , even irrational roots occur in pairs. Hence if 2+root(3) is a root , then even 2-root(3) is a root .
(All these are very useful in finding number of positive , negative , real ,complex etc roots of an equation )

[gkathotia]

Read root(3) as the square root of 3 .

Gaurav

[pagalguy]

Actually there is my personal gain from this section.

Fine fine , but I still think otherwise. It takes a lot of effort to do what you are doing here. This thread is being edited to be shown as a sticky.

Regards,