Question

$x^{(3/4)(\log_{2}x)^{2} + \log_{2}x - 5/4}$ = $\sqrt{2}$ has –

• A. Exactly two real roots
• B. No real root
• C. One irrational root
• D. None of these

Answer 1

Answer: (C)

One irrational root

Answer 2

Taking log of the sides in (1), we get

Ž $\left\lbrack \frac{3}{4}(\log_{2}x)^{2} + \log_{2}x - \frac{5}{4} \right\rbrack$log₂ x = log₂ ($\sqrt{2}$)

Ž $\left( \frac{3}{4}y^{2} + y - \frac{5}{4} \right)$y = $\frac{1}{2}$log₂ 2 where y = log₂ x

Ž 3y³ + 4y² – 5y – 2 = 0

Ž 3y³ – 3y² + 7y² – 7y + 2y – 2 = 0

Ž (y – 1) (3y² + 7y + 2) = 0

Ž (y – 1) [3y² + 6y + y + 2] = 0

(y – 1) (y + 2) (3y + 1) = 0

Ž y = 1, – 2, – 1/3

Ž log₂ x = 1, – 2, – 1/3 Ž x = 2, $\frac{1}{4}$, 2^–1/3

Thus, (1) has one irrational root viz. 2^–1/3.