Question

The equation $x^{\frac{3}{4}({\log }_{2}x{)}^2+{\log }_{2}x-\frac{5}{4}}=\sqrt{2}$ has

• A. atleast one real solution
• B. exactly three real solutions
• C. exactly one irrational solution
• D. complex roots

Answer 1

Answer: (A), (B), (C)

Given, $x^{\frac{3}{4}({\log }_{2}x{)}^2+{\log }_{2}x-\frac{5}{4}}=\sqrt{2}$
$\Rightarrow \frac{3}{4}({\log }_{2}x{)}^2+{\log }_{2}x-\frac{5}{4}={\log }_x\sqrt{2}$
$\Rightarrow \frac{3}{4}({\log }_{2}x{)}^2+{\log }_{2}x-\frac{5}{4}=\frac{1}{2{\log }_{2}x}$
$\Rightarrow 3({\log }_{2}x{)}^3+4({\log }_{2}x{)}^2-5({\log }_{2}x)-2=0$
Put ${\log }_{2}x=y$
$\therefore 3y^3+4y^2-5y-2=0$
$\Rightarrow (y-1)(y+2)(3y+1)=0$
$\Rightarrow y=1,-2,-1/3$
$\Rightarrow {\log }_{2}x=1,-2,-1/3$
$\Rightarrow x=2,\frac{1}{2^{1/3}},\frac{1}{4}$