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The equation x3/4(log2x)^2 + log2x - 5/...

The equation x3/4(log2x)^2 + log2x - 5/4 = √2 has (a) at least one real solution (b) exactly three real solutions (c) exactly one irrational solution (d) complex roots

A

altest one real solution

B

exactly three real solutions

C

exactly one irrational solution

D

complex roots

Text Solution

Verified by Experts

The correct Answer is:
B
Given `x^(3/4(log_(2)x)^(2)+log^(2)+log_(2)x-5/4)=sqrt2`
`implies3/4(log_(2)x)^(2)+log_(2)x-5/4=log_(x)sqrt2`
`implies3/4(log_(2)x)^(2)+log_(2)x-5/4=(1)/(2log_(2)x)`
`implies3(log_(2)x)^(3)+4(log_(2)x)^(2)-5 (log_(2)x)-2=0`
Put `log_(2)x=y`
`therefore3y^(3)+4y^(2)-5y-2=0`
`implies(y-1)(y+2)(3y+1)=0`
`impliesy=1,-2,-1//3`
`implieslog_(2)x=1,-2,-1//3`
`impliesx=2, (1)/(2^(1//3)),1/4`
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