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The equation \[\frac{x}{x+1} + \frac{x}{x+2} = kx\]has exactly two complex roots. Find all possible complex values for $k.$  Enter all the possible values, separated by commas.

 Dec 27, 2020
 #1
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Well I divided it by $x$, after doing that I got \[\frac{1}{x+1} + \frac{1}{x+2} = k\], doing this i equalized the denominator resulting in this:

 

\[\frac{2x+3}{(x+1)(x+2)} = k\]

 Dec 27, 2020
 #2
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Now I am stuck

 Dec 27, 2020
 #3
avatar+117724 
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Multiply your equation  through  by    (x + 1) ( x + 2)

 

And we have that

 

2x + 3  =   k ( x^2  + 3x + 2)

 

2x + 3  =  kx^2  + 3kx  + 2k

 

kx^2   + (3k - 2)x  + (2k  -3)  =  0

 

For this to have complex roots.....the discriminant must  be <  0

 

So....

 

(3k - 2)^2  - 4 ( k) ( 2k -3)  <  0

 

9k^2  - 12k + 4   - 8k^2 + 12k  <  0

 

k^2  +  4   <  0    (1)

 

k^2  <  -4      take  both roots

 

k <  -2i         k > 2i

 

Any  value of  k < -2i  produces complex roots

And any value of k > 2i  produces  complex roots

 

 

cool cool cool

 

 

 

 

 

 

 

,    

 Dec 27, 2020
 #4
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Hi, Cphill this says for you to put this in an format of separated commas, I think your answer is incorrect but I'm not sure, since the question asks for commas, the only possible way to submit your answer is interval notation. The question gives the formatting tips and never says interval notation. I actually got the same answer, the only difference of out strategies is that at the start I subtracted $k$ but you cross multiplied and we got the same answer, and yes, I have been spending hours on this problem,

 Dec 27, 2020

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