SOLUTION: Given the equation kx^2+(k-1)x+2k+3=0 where k is a non-zero integer, find the value of k for which: (a) one root is the negative of the other. (b) one root is the reciprocal of

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Given the equation kx^2+(k-1)x+2k+3=0 where k is a non-zero integer, find the value of k for which: (a) one root is the negative of the other. (b) one root is the reciprocal of       Log On


   

Question 173793: Given the equation kx^2+(k-1)x+2k+3=0 where k is a non-zero integer, find the value of k for which:
(a) one root is the negative of the other.
(b) one root is the reciprocal of the other.
(c) one root is the twice of the other.
Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
If I say the roots are r%5B1%5D and r%5B2%5D, then

given the equation
kx%5E2+%2B+%28k-1%29x+%2B+2k+%2B+3+=+0
divide both sides by k
x%5E2+%2B+%28%28k-1%29%2Fk%29%2Ax+%2B+%282k+%2B+3%29%2Fk+=+0
-----------------------------------------
(a) one root is the negative of the other.
r%5B1%5D+=+-r%5B2%5D
r%5B1%5D+%2B+r%5B2%5D+=+0
therefore the coefficient of x is zero
%28k+-+1%29%2Fk+=+0
k+=+1
------------------------------------------
r%5B1%5D+=+1%2Fr%5B2%5D
r%5B1%5D%2Ar%5B2%5D+=+1
%282k%2B+3%29%2Fk+=+1
2k+%2B+3+=+k
k+=+-3
-------------------------------------------
(c) one root is the twice of the other.
r%5B1%5D+=+2%2Ar%5B2%5D
-%28r%5B1%5D+%2B+r%5B2%5D%29+=+-3%2Ar%5B2%5D
-3%2Ar%5B2%5D+=+%28k+-+1%29%2Fk
r%5B2%5D+=+%281+-+k%29%2F%283k%29
r%5B1%5D%2Ar%5B2%5D+=+2%2A%28r%5B2%5D%29%5E2
2%2A%28r%5B2%5D%29%5E2+=+%282k+%2B+3%29%2Fk
2%2A%28%281+-+k%29%2F%283k%29%29%5E2+=+%282k+%2B+3%29%2Fk
%284%2A%281-k%29%5E2%29%2F+%289k%5E2%29+=+%282k+%2B3%29%2Fk
4k%2A%281-k%29%5E2+=+9k%5E2%2A%282k+%2B+3%29
4k%2A%281+-+2k+%2B+k%5E2%29+=+18k%5E3+%2B+27k%5E2
4k+-+2k%5E2+%2B+4k%5E3+=+18k%5E3+%2B+27k%5E2
14k%5E3+%2B+29k%5E2+-+4k+=+0
k%2A%2814k%5E2+%2B+29k+-+4%29+=+0
k+=+0 and the solution to the quadratic
(unless I goofed)