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Manipulating Roots


Date: 08/08/98 at 12:46:32
From: Vivian Wave
Subject: Equations and Functions

Hello:

It would be great if you could help me to solve this problem. 

If a and b are the roots of the equation 3x^2 - 5x + 1 = 0, find the 
equation whose roots are a/b and b/a.

I appreciate your assistance. Thank you.


Date: 08/10/98 at 12:54:54
From: Doctor Peterson
Subject: Re: Equations and Functions

Hi, Vivian. 

This is an interesting question. You could, of course, just solve the 
original quadratic and then write an equation whose roots are a/b and 
b/a, but there's a nice trick that lets you do it more quickly.

The basic idea is that if you have an equation of the form:

   A*x^2 + B*x + C = 0

and its two roots are a and b, then the equation can also be written 
as:

   A*(x - a)*(x - b) = 0

(where I've put in the factor A so the coefficient of x^2 will match 
the original equation), and this expands to:

   A*x^2 - A*(a+b)*x + A*a*b = 0

This means that:

   a+b = -B/A   and
   a*b = C/A

In your case, A = 3, B = -5, and C = 1, so we have:

    a+b = 5/3     and
    a*b = 1/3

Now see if you can reverse this and find what A, B, and C have to be 
(you can take A = 1) if the roots are a/b and b/a.

(Hint: the product of the roots of this new equation will be

    a   b
    - * - = 1
    b   a

and the sum of the roots will be

    a   b   b + a
    - + - = -----
    b   a   b * a

and you can figure these out easily.)

- Doctor Peterson, The Math Forum
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Associated Topics:
High School Basic Algebra
High School Equations, Graphs, Translations
High School Functions

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