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luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

simultaneous equations related to cubic equation roots

i've been working through one of my maths textbooks for one of the exams im taking in january, and ive had to skip a small section because it honestly has me a bit stumped.

it has gone through first for quadratics, that for some quadratic ax² + bx + c = 0, with roots alpha, beta, the following are true

and some useful identities like:

so that you can use these to find a quadratic equation with a given couple of roots, or from one equation find another equation satisfying some couple of roots that are symmetric to the others, like quadratic equation 2x^2 - 3x + 5 = 0 has roots a,b find an equation with roots 2/a 2/b etc.

and then thus its quite simple to solve a set of simultaneous equations in the form for example a + b = 10, ab = 20

it then goes onto cubics

and some other identities, but im stuck on solving 3 sim. equations relating to this whole thing.

here is an example question on this:

i can do part (a) no problem but i can't figure out a way of playing with the equations to solve the equations for p,q,r

Last edited by luca-deltodesco (2007-10-26 09:10:20)

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luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: simultaneous equations related to cubic equation roots

here is my working out:

and from here im stuck on where to go further

all that i can guess is that i can now formulate a cubic equation with roots p,q,r and solve that, but i'm not sure if that is how i'm meant to go about it?

--------

this is a bit of a tangent, but the book hasn't given any guidance on solving cubic equations, am i to assume that im simply meant to try and guess for the first root to gain an easily solvable quadratic? for example here i can see that one root is 1, but would i be given credit for simply writing that one root is 1 and showing it works?

going along that lines anyway i get:

which is correct according to answers in book, but would that be a fine way to answer it? (yes i realised i started with a question, then answered it with another question, then half answered that )

also, i've written there that p,q,r are members of the set -1,1,2 such that they are none equal, is that a valid way of stating it? the book has written "p,q,r can be any permutation of -1,1,2"

Last edited by luca-deltodesco (2007-10-26 09:26:02)

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The Beginning Of All Things To End.
The End Of All Things To Come.

blegh123

Guest

Re: simultaneous equations related to cubic equation roots

here is my working out:

and from here im stuck on where to go further

all that i can guess is that i can now formulate a cubic equation with roots p,q,r and solve that, but i'm not sure if that is how i'm meant to go about it?

--------

this is a bit of a tangent, but the book hasn't given any guidance on solving cubic equations, am i to assume that im simply meant to try and guess for the first root to gain an easily solvable quadratic? for example here i can see that one root is 1, but would i be given credit for simply writing that one root is 1 and showing it works?

going along that lines anyway i get:

which is correct according to answers in book, but would that be a fine way to answer it? (yes i realised i started with a question, then answered it with another question, then half answered that )

also, i've written there that p,q,r are members of the set -1,1,2 such that they are none equal, is that a valid way of stating it? the book has written "p,q,r can be any permutation of -1,1,2"

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: simultaneous equations related to cubic equation roots

why did you just quote my post on its own?

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The Beginning Of All Things To End.
The End Of All Things To Come.