a.) The roots of the equation ax^2 + bx + c =0 , where a belongs to real number , are two consecutive odd positive integers , then 1.) |b|=4a 3.) |b|=4a 4.) None of these Answer : 2 b.) The roots of the equation (3-x)^4 + (2-x)^4 = (5-2x)^4 are 1.) Two Real two imaginary 2.) All imaginary 3.) All Real 4.) None of these Answer : 1

Question

a.) The roots of the equation ax^2 + bx + c =0 , where a belongs to real number , are two consecutive odd positive integers , then 
1.) |b|<=4a 
2.) |b|>=4a
3.) |b|=4a
4.) None of these 
Answer : 2

b.) The roots of the equation (3-x)^4 + (2-x)^4 = (5-2x)^4 are 
1.) Two Real two imaginary 
2.) All imaginary
3.) All Real
4.) None of these
Answer : 1

anonymous · Answer

a) The roots are x1 = 2 k - 1 and  x2 = 2 k + 1 for some natural k.
x1 + x2 = |b|/a => 4k*a = |b| => |b|>= 4*a  

b) Denote 3-x = a, 2-x = b => 5-2x = a + b
or
a^4 + b^4 = (a + b ) ^4
or after simplification:
2 a b ( 2 a^2 + 3 a b + 2 b^2 ) = 0  
which gives two real roots 2 and 3. Expression in parenthesis after substitution gives equation 7 x^2 -35 x + 44 = 0 , which yields other two complex roots.