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Roots and coeffecients of polynomial equations

  • Thread starter olee
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  • #1
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Homework Statement


One root of 2x2-kx+k=0 is twice the other. Find K. (assume k is not 0)


Homework Equations


not sure what this means.


The Attempt at a Solution



[tex]\alpha[/tex]+[tex]\beta[/tex]=[tex]\frac{-b}{a}[/tex]

[tex]\alpha\beta[/tex]=[tex]\frac{c}{a}[/tex]

then i tried

[tex]\alpha[/tex]=2[tex]\beta[/tex] then substituted in..

2x2-3[tex]\beta[/tex]+[tex]\alpha\beta[/tex]=0 and now im stuck

im not sure if this is right way to do this
 

Answers and Replies

  • #2
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Let r and 2r be the two roots. Substitute each one in your equation. That will give you two equations in two unknowns - r and k. Solve the two equations simulataneously for these variables.
 
  • #3
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Let r and 2r be the two roots. Substitute each one in your equation. That will give you two equations in two unknowns - r and k. Solve the two equations simulataneously for these variables.
thanks for reply. which equation are you mentioning?
 
  • #4
33,162
4,847
The only equation you were given, and the one whose roots you are trying to find.
2x2 - kx + k=0
 
  • #5
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The only equation you were given, and the one whose roots you are trying to find.
2x2 - kx + k=0
thanks for reply. but i still have two unknowns and i can't solve

edit: and i still have the x at the start of that equation. all i know is 'a' (which is 2)
 
  • #6
33,162
4,847
You should have two equations in the two unknowns. Solve these equation simultaneously to find r and k.
 
  • #7
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You should have two equations in the two unknowns. Solve these equation simultaneously to find r and k.
by substituting in, do you mean where the -k is (which is b) i substitute in 2r+4r? (to get this i had r+2r = -[tex]\frac{b}{2}[/tex])

and then where the +k is (which is c) i substitute in 4r2? (to get this i rearranged 2r2=[tex]\frac{c}{2}[/tex]
 
  • #8
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by substituting in, do you mean where the -k is (which is b) i substitute in 2r+4r? (to get this i had r+2r = -[tex]\frac{b}{2}[/tex] )

and then where the +k is (which is c) i substitute in 4r2? (to get this i rearranged 2r2=[tex]\frac{c}{2}[/tex]
edit: was to remote the extra bracket
 
  • #9
33,162
4,847
What I'm saying is that you should ignore the equations you have with alpha and beta and a and b, and focus on the equation you're trying to solve, 2x2 - kx + k = 0.

You are given that r and 2r are roots of this equation, so if you replace x by r you'll get one equation, and replace x by 2r, you'll get another equation. These two equations will be in terms of r and k.

Solve those two equations simultaneously for r and k.
 
  • #10
epenguin
Homework Helper
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You got the equation

2x2-3[tex]\beta[/tex]+[tex]\alpha\beta[/tex]=0

In the 2nd term here you have used the information you are given (one root is twice the other) and in the third term you haven't yet.

Oh, and you must have meant

2x2- 3[tex]\beta[/tex] x + [tex]\alpha\beta[/tex]=0
 
Last edited:
  • #11
24
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What I'm saying is that you should ignore the equations you have with alpha and beta and a and b, and focus on the equation you're trying to solve, 2x2 - kx + k = 0.

You are given that r and 2r are roots of this equation, so if you replace x by r you'll get one equation, and replace x by 2r, you'll get another equation. These two equations will be in terms of r and k.

Solve those two equations simultaneously for r and k.
thanks for reply. doing this way i made the two equations equal eachother (2r2-kr+k = 8r2-2kr+k
ended up with r=[tex]\frac{6}{k}[/tex] and i can't solve that because of two unknown variables.




You got the equation

2x2-3[tex]\beta[/tex]+[tex]\alpha\beta[/tex]=0

In the 2nd term here you have used the information you are given (one root is twice the other) and in the third term you haven't yet.

Oh, and you must have meant

2x2- 3[tex]\beta[/tex] x + [tex]\alpha\beta[/tex]=0

thanks for reply. this method i ended up with 2x2-3[tex]\beta[/tex]x+2[tex]\beta[/tex]2
 
  • #12
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4,847
I get r = k/6 (not 6/k) or r = 0.

If r = 0, then 2r = 0, so what is k?
If r = k/6, then 2r = k/3, so what is k?
 
  • #13
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I get r = k/6 (not 6/k) or r = 0.

If r = 0, then 2r = 0, so what is k?
If r = k/6, then 2r = k/3, so what is k?
thanks for reply. yeah i got r=k/6 (i mistyped)

If r = 0, then 2r = 0, so what is k? k=0
If r = k/6, then 2r = k/3, so what is k? plugged r=k/6 back into original equation and got k = 9 or 0.. right?

EDIT: made a mistake, recalculated and got 9 and 0
 
  • #14
epenguin
Homework Helper
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thanks for reply. doing this way i made the two equations equal eachother (2r2-kr+k = 8r2-2kr+k
ended up with r=[tex]\frac{6}{k}[/tex] and i can't solve that because of two unknown variables.







thanks for reply. this method i ended up with 2x2-3[tex]\beta[/tex]x+2[tex]\beta[/tex]2
Sorry -I made a typo.

It is usually less confusing to work with an equation whose leading coefficient is 1. So better to divide your initial equation by 2. So use

x2 - kx/2 + k/2

A quadratic with one root twice the other is

x2-3[tex]\beta[/tex]x+2[tex]\beta[/tex]2

I think you had the idea of that but the leading coefficient caused you confusion. You should easily be able to see your last result must be wrong; do not accept any reult without checking it against your original problem.
 
Last edited:
  • #15
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Sorry -I made a typo.

It is usually less confusing to work with an equation whose leading coefficient is 1. So better to divide your initial equation by 2. So use

x2 - kx/2 + k/2

A quadratic with one root twice the other is

x2-3[tex]\beta[/tex]x+2[tex]\beta[/tex]2

I think you had the idea of that but the leading coefficient caused you confusion. You should easily be able to see your last result must be wrong; do not accept any reult without checking it against your original problem.
so how do i solve it from here? im really confused..
 
  • #16
epenguin
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Any, in other words every, quadratic with one root twice the other is of form

[tex]x^2-3\beta x+2\beta^2[/tex] (1)

(a result you got but for a little mistake.)

You are told

[tex]x^2 - \frac{kx}{2} + \frac{k}{2} [/tex] (2)

is a quadratic with one root twice the other.

So (2) must be of form (1). Or if you like identical to it for all x. So do you see how to compare them (remembering similar exercises you've probably done before for identical expressions)?
 
Last edited:

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