Cubic equation with two unknown coefficients

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SUMMARY

The discussion centers on solving the cubic equation x^3 + px^2 + 56x + q = 0, where the roots are positive and form a geometric series with a common ratio of 2. Participants aim to determine the roots, as well as the coefficients p and q. The key relationships established are that the sum of the roots equals -p, the sum of the products of the roots taken two at a time equals 56, and the product of the roots equals -q. Understanding the properties of geometric sequences is crucial for finding the roots and coefficients.

PREREQUISITES
  • Understanding of cubic equations and their roots
  • Knowledge of geometric sequences and their properties
  • Familiarity with Vieta's formulas for polynomial equations
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the properties of geometric sequences, particularly with a common ratio
  • Learn how to apply Vieta's formulas to find polynomial coefficients
  • Explore methods for solving cubic equations analytically
  • Practice problems involving roots of polynomials and their relationships
USEFUL FOR

Students studying algebra, particularly those tackling polynomial equations and geometric sequences, as well as educators looking for examples to illustrate these concepts.

MegaDeth
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1. Homework Statement

x^3 +px^2 + 56x + q = 0


I'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.

2. 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.


3. 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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The most important piece of information here is that the three roots are part of a geometric sequence with factor 2.

What is a geometric sequence with factor 2?? What is the formula?? How can you express an element with respect to a previous element??
 
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