155. Prove that the equation \( (b x+c)^{2}-a^{2} x^{4}=0 \) has a least two real roots for all \( a, b, c \in R \).

Question

Prove that the equation \( (b x+c)^{2}-a^{2} x^{4}=0 \) has a least two real roots for all \( a, b, c \in R \).

Answer 1

( b x + c )² − a² x⁴  = 0

(b x + c + ax² )( b x + c - ax² ) = 0

for  ( ax² + b x + c +) = 0 

it would have real roots if  b² - 4ac >0

for  ( - ax² + b x + c +) = 0 

it would have real roots if  b² + 4ac >0

Case 1 : if  a >=0 and c>= 0

             b² + 4ac >0 then it will have 2 real roots

Case 2 : if  a < 0 and c< 0

             b² + 4ac >0 then it will have 2 real roots

Case 3 : if  a < 0 and c >=  0

              b² - 4ac >0 then it will have 2 real roots

Case 4 : if  a >= 0 and c <  0

            b² - 4ac >0 then it will have 2 real roots