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.
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:
\[\frac{2x+3}{(x+1)(x+2)} = k\]
+128473 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
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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,
