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shubhamsrg
Group Title
TUTORIAL :
on how to solve a 3 degree eqn in x using trigonometry .. i learnt this method quite some time ago and maybe everyone should give this a reading..
 one year ago
 one year ago
shubhamsrg Group Title
TUTORIAL : on how to solve a 3 degree eqn in x using trigonometry .. i learnt this method quite some time ago and maybe everyone should give this a reading..
 one year ago
 one year ago

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shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
we have : ax^3 + bx^2 + cx+ d =0 for simplification, i like to keep coefficient of x^3 = 1 for that,,lets divide both sides by a (given a not equal to 0) lets say b/a =p , c/a = q , d/a =r => x^3 + px^2 + qx +r =0 what we aim here firstly is to convert this into depressed form ( make coefficient of 2nd highest degree =0) lets substitute x = (y + m) , for some new variable y and a constant m , we will understand the purpose of doing this soon.. => (y+m)^3 + p(y+m)^2 + q(y+m) + r =0 =>y^3 + y^2(p + 3m) + y(2m + 3m^2 +q) + (m^3 + pm^2 + qm +r) =0 as i said earlier,we want p+ 3m =0 => m= p/3 thus on making that substitution, we'll get y^3 + Ay + B = 0 where A and B are some known constants.. now if we calculate y , we can easily calculate x as x= y+m or x= y p/3 here comes the trignometric part: this glorious formulla will do wonders for us as you'll find below : sin 3@ = 3sin @  4sin^3 @ (@ denotes theta) so we have y^3 + Ay + B=0 let y = t sin@ for some constant t , => t^3 sin^3 @ + At sin@ + B =0 lets multiply both sides by 4/t^3 => 4 sin^3 @ + 4A sin@ /t^2 + 4B/t^3 = 0 we have to chose t such that coefficient of sin@ = 3 => 4A/t^2 = 3 =>t = sqrt( 4A/3) and sqrt( 4A/3) on making that substitution, we see we have 4sin^3 @ + 3sin@ =C for some new constant C. => sin3@ = C =>3@ = sin^1 (C) @ = 1/3 * sin^1 (C) since you know the value of @, you then know value of y = t sin@ and then x = y+m
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
all this might seem tedious and long at first,,but this is very advantageous to know.. this one yields both complex roots as well as real roots..2 of them .. (since you get 2 values of t) hope that helps..
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
another thing which we can make out from here that, in any eqn like x^n + px^(n1) + qx^(n2) ......... + constant term =0 to make it depressed eqn , we can always substitute x = y  (p/n)
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
What is this method called ?
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
i dont know/remember.. :
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
3x^3+2x^2+x+1=0 Show me how to solve this step by step please.
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
you do it and i shall add my words in between when required..
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
let me start then :)
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
Do i have to suppose x^3=1 ?
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
nops..i meant coefficient of x^3 should be made 1,,just for simplification.. so you may divide both sides by 3
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
Dividing the whole equation by 3 x^3+2/3 x^2+1/3x+1/3=0
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
x = y  p/3 .. make that substitution,, p = 2/3 here..
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
p=2/3,q=1/3,r=1/3 I did not get the substitution part lol ? Do you think this method is more effective then the others ?
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
i cant say that,,since i dont know other methods! :P nevermind,,whats the problem is substituting ?
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
like in x+2 = y, you can substitute directly x=2 ,, similarly,,substitute x= y  p/3
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
i'll give a kick.. x^3+2/3 x^2+1/3x+1/3=0 (y  2/9)^3+ 2/3 (y 2/9)^3 + 1/3 ( y 2/9) + 1/3 = 0 did you follow?
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
Ok so we have just done a simple substitution here x=y2/9.
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
Do i expand them using formula ?
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
yep..expand and simplify .. write what you get .. sorry am too lazy to check your work or do it myself..so be careful..please dont mind. :P
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
Anyways i understood the part where i was confused,Thanks a lot.
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
so,,you're comfortable with the rest ?
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
Yeah i was confused in this part lol :D
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
kk.glad i could help..
 one year ago

Hero Group TitleBest ResponseYou've already chosen the best response.0
Is there a textbook or video tutorial I can find this in?
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
If i remember the name of the method i could have told you :(
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
well one of my seniors had told me this..i really cant tell you any name or any vids or theories about it..sorry.. anyways,,are you stuck somewhere ?
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
@Hero This method does not intrest me :) Its very long.
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
sorry buddy,,too lazy doing all that..though i could help you ..but giving examples..nah,,not gonna do that..
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
ZZZZZzzz...
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
@shubhamsrg Please.
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
i dont regret that..i made an effort..and it was satisfactory for me..i dont mind your foolish reviews since you ofcorse,dont get it..
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
@shubhamsrg If you give an example maybe you would be getting more medals think about that :D
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
ohh yes,,the medals..it was always my dream to earn more and more medals on open study.. !! o.O
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
Oh come on just provide us with an example please ?
 one year ago

hba Group TitleBest ResponseYou've already chosen the best response.1
If you feel lazy do it after sometime :)
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
nevermind,,not gonna do that..if you dont get it..put up your query here..i may help/improvise// but i hate doing hard work more often than not..
 one year ago

RajshikharGupta Group TitleBest ResponseYou've already chosen the best response.0
it is tooo lengthy but indeed beautiful we use to call it 'MAJDOORI'
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
we?? o.O who all do you represent? :P
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
alright,,seems its lengthy.. in that case,,gimme some time,,i'll post a general formula to the value of x directly..
 one year ago

sirm3d Group TitleBest ResponseYou've already chosen the best response.0
except for the trigonometric substitution, solving a cubic equation appeared in Ars Magna of Girolamo Cardamo, or Cardan.
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
here's the general formula for solution of x in x^3 + px^2 + qx + r =0 using trig. substitution : dw:1354008782695:dw
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
where: m = sqrt(4A /3) and sqrt(4A/3) A = q (p^2 /3) B = (2p^3 /27) + (qp /3) + r
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
hope that aids to the lengthiness ..
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
sorry, correction : that'll be p/3 not +p/3 there in the drawing.. rest all the same..
 one year ago

inkyvoyd Group TitleBest ResponseYou've already chosen the best response.0
cardano is better. except casus irriduciblis. Then this is.
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
ohh..i just googled over!! that was amazing! thanks @inkyvoyd also,,just found out vieta's substitution method..too good..probably the easiest..
 one year ago

Mathmuse Group TitleBest ResponseYou've already chosen the best response.0
Haha, reading about this and found this excerpt: The longsought solution of the general cubic was found, in 1535 by Niccol`o Tartaglia (c. 150057). This achievement brought him great celebrity, and he spent the next 10 years visiting the crownedheads of Europe and solving their cubics for them. However, he was persuaded to divulge his secret, on the promise of complete conﬁdentiality, by Girolamo Cardano (1501–76), who promptly published it in his algebra book The Great Art. There followed an acrimonious dispute between Tartaglia and Cardano which preoccupied much of Tartaglia’s later life. At one point, the two agreed to a public duel, in which they would each bring along their favourite mathematical problems for the other to solve. However, it never took place @Hero http://www.admissionstests.cambridgeassessment.org.uk/adt/digitalAssets/110501_Advanced_Problems_in_Mathematics.pdf question #12 has an example of this.
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
that was interesting @mathmuse ..thanks for sharing
 one year ago

mukushla Group TitleBest ResponseYou've already chosen the best response.0
nice one :)
 one year ago

shubhamsrg Group TitleBest ResponseYou've already chosen the best response.12
aha..thanks..your appreciations means a lot to me sir.. :)
 one year ago
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