Cubic equation with two unknown coefficients

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Homework Help Overview

The problem involves a cubic equation of the form x^3 + px^2 + 56x + q = 0, where the roots are positive and form the first three terms of a geometric series with a common ratio of 2. Participants are tasked with finding the roots as well as the coefficients p and q.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the nature of the roots, suggesting they could be expressed in terms of a variable r, with some proposing specific forms like r, r+2, and r+4, while others suggest r, 2r, and 4r. There are questions about how to proceed to find p and q using these roots.

Discussion Status

The discussion is ongoing, with participants exploring different interpretations of the roots and their relationships. Some guidance has been offered regarding the expressions for the roots, but there is a lack of clarity on the method to find the coefficients p and q.

Contextual Notes

Participants express varying levels of familiarity with the methods being discussed, indicating potential gaps in understanding the approach to solving the problem.

MegaDeth
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Homework Statement



x^3 +px^2 + 56x + q = 0I've attempted the first part but I have no idea what to do next. I know usually you'd have 3 new roots inn terms of alpha, beta and gamma but they're not given.

Homework Equations



Given that the three roots are all positive and are the first free terms of a geometric series with common ratio 2,

a. find the three roots of the equation.

b. find the values of p and q.

The Attempt at a Solution



I've work out the old roots,

roots one at a time = -p

roots two at a time = 56

product = -q
 
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Hi MegaDeth! :smile:
MegaDeth said:
Given that the three roots are all positive and are the first free terms of a geometric series with common ratio 2 …

(i assume you mean the first three terms? :wink:)

start by calling the smallest root "r" …

then you know what the other two roots are :wink:​
 
tiny-tim said:
Hi MegaDeth! :smile:


(i assume you mean the first three terms? :wink:)

start by calling the smallest root "r" …

then you know what the other two roots are :wink:​


Oh god, that's embarrassing. I didn't notice that mistake.

Ok, so the roots would be: r, r+2 and r+4? If so, how would I use these to find p and q?
 
MegaDeth said:
Ok, so the roots would be: r, r+2 and r+4? If so, how would I use these to find p and q?

r, 2r, 4r :wink:

try it, then ask :smile:​
 
Sorry but I'm not familiar with this method. I don't know how to do it. :(
 
well, write out the equations …

what are they?​
 
MegaDeth said:
Sorry but I'm not familiar with this method. I don't know how to do it. :(
What is the method that you're unfamiliar with ?
 

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