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 2 years ago
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..
 2 years ago
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..

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shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12we 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

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12all 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..

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12another 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)

hba
 2 years ago
Best ResponseYou've already chosen the best response.1What is this method called ?

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12i dont know/remember.. :

hba
 2 years ago
Best ResponseYou've already chosen the best response.13x^3+2x^2+x+1=0 Show me how to solve this step by step please.

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12you do it and i shall add my words in between when required..

hba
 2 years ago
Best ResponseYou've already chosen the best response.1Do i have to suppose x^3=1 ?

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

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

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

hba
 2 years ago
Best ResponseYou've already chosen the best response.1p=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 ?

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

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

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12i'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?

hba
 2 years ago
Best ResponseYou've already chosen the best response.1Ok so we have just done a simple substitution here x=y2/9.

hba
 2 years ago
Best ResponseYou've already chosen the best response.1Do i expand them using formula ?

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12yep..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

hba
 2 years ago
Best ResponseYou've already chosen the best response.1Anyways i understood the part where i was confused,Thanks a lot.

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12so,,you're comfortable with the rest ?

hba
 2 years ago
Best ResponseYou've already chosen the best response.1Yeah i was confused in this part lol :D

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12kk.glad i could help..

Hero
 2 years ago
Best ResponseYou've already chosen the best response.0Is there a textbook or video tutorial I can find this in?

hba
 2 years ago
Best ResponseYou've already chosen the best response.1If i remember the name of the method i could have told you :(

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12well 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 ?

hba
 2 years ago
Best ResponseYou've already chosen the best response.1@Hero This method does not intrest me :) Its very long.

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

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12i 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..

hba
 2 years ago
Best 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

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

hba
 2 years ago
Best ResponseYou've already chosen the best response.1Oh come on just provide us with an example please ?

hba
 2 years ago
Best ResponseYou've already chosen the best response.1If you feel lazy do it after sometime :)

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12nevermind,,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..

RajshikharGupta
 2 years ago
Best ResponseYou've already chosen the best response.0it is tooo lengthy but indeed beautiful we use to call it 'MAJDOORI'

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12we?? o.O who all do you represent? :P

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

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

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

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

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12hope that aids to the lengthiness ..

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

inkyvoyd
 2 years ago
Best ResponseYou've already chosen the best response.0cardano is better. except casus irriduciblis. Then this is.

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

Mathmuse
 2 years ago
Best ResponseYou've already chosen the best response.0Haha, 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.

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12that was interesting @mathmuse ..thanks for sharing

shubhamsrg
 2 years ago
Best ResponseYou've already chosen the best response.12aha..thanks..your appreciations means a lot to me sir.. :)
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