If the equation 16 x 4 − m x 3 + ( 2 m + 17 ) x 2 − m x + 16 = 0 has four distinct roots forming a geometric progression, and one of the roots is 2 1 , then:

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If the equation 16 x 4 − m x 3 + ( 2 m + 17 ) x 2 − m x + 16 = 0 has four distinct roots forming a geometric progression, and one of the roots is 2 1 ​ , then:

TestHub · Accepted Answer

Putting x = 2 1 ​ : 1 − 8 m ​ + 4 2 m + 17 ​ − 2 m ​ + 16 = 0 ⇒ m = 170 . The equation becomes 16 x 4 − 170 x 3 + 357 x 2 − 170 x + 16 = 0 , a reciprocal equation with roots 8 1 ​ , 2 1 ​ , 2 , 8 , which form a G.P. with ratio 4. Sum of roots = 16 170 ​ = 8 85 ​ . Product of roots = 16 16 ​ = 1 , not 3.

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