A) Real and distinct
B) Real and equal
C) Imaginary
D) None of these
Correct Answer: A
Solution :
Given equation can be rewritten as \[3{{x}^{2}}-(a+c+2b+2d)x+(ac+2bd)=0\] Its discriminant D \[={{(a+c+2b+2d)}^{2}}-4.3(ac+2bd)\] \[={{\left\{ (a+2d)+(c+2b) \right\}}^{2}}-12(ac+2bd)\] \[={{\left\{ (a+2d)-(c+2b) \right\}}^{2}}+4(a+2d)(c+2b)-12(ac+2bd)\] \[={{\left\{ (a+2d)-(c+2b) \right\}}^{2}}-8ac+8ab+8dc-8bd\] \[={{\left\{ (a+2d)-(c+2b) \right\}}^{2}}+8(c-b)(d-a)\] which is +ve, since \[a<b<c<d\]. Hence roots are real and distinct.
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