IIT JEE , AIEEE Forum - Mathematics , Algebra

neeraj_agarwal_1990

The equation ${x}^{\frac{3}{4}[{{log}_{2}x}]^{2}+{log}_{2}x-\frac{5}{4}}=\sqrt{2}$ has
a)at least 1 real solution
b)at least 3 real solutions
c)exactly one real solution
d)All of these

ans-c
i am getting x= 2 and ${2}^{\frac{1}{3}}$ as the solutions of this equation (taking log on both sides)

Conjurer

Re:triplets

How do I use LATEX here?

Obviously there cannot be 2 real roots(it may have 3) to this equation as it is cubic.

neeraj_agarwal_1990

i am getting x=2 as a double root....

RyuAmakusa

there are 3 roots 2, 2^-1/3, 1/4

RyuAmakusa

take log with base 2 on both sides. let log₂ x = t on taking log u get (3/4t² + t - 5/4)t = 1/2 => 3t³ +4 t² -5t -2 =0. => 3t³ +6 t²-2 t² -4t -t -2 =0. =>3t²(t+2) -2t(t+2) -(t+2) =0 =>(t+2)(3t² - 2t -1)=0 =>(t+2)(3t+1)(t -1)=0 => t = 1,-1/3,-2. so log₂ x =1,-1/3,-2. => x= 2,2^-1/3,2^-2. so 3 roots.

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