The number of solutions of the equation is : [2025]

Question

The number of solutions of the equation $(\frac{9}{x} - \frac{9}{\sqrt{x}} + 2) (\frac{2}{x} - \frac{7}{\sqrt{x}} + 3) = 0$ is : [2025]

Smart Achievers · Accepted Answer

(3)

We have, $(\frac{9}{x} - \frac{9}{\sqrt{x}} + 2) (\frac{2}{x} - \frac{7}{\sqrt{x}} + 3) = 0$ ... (i)

Put $\frac{1}{\sqrt{x}} = y$ in (i), we get

$(9 y^{2} - 9 y + 2) (2 y^{2} - 7 y + 3) = 0$

$\Rightarrow (3 y - 2) (3 y - 1) (2 y - 1) (y - 3) = 0$

$\Rightarrow y = \frac{2}{3}, \frac{1}{3}, \frac{1}{2}, 3 \Rightarrow x = \frac{9}{4}, 9, 4, \frac{1}{9}$

$∴$ Number of solutions = 4.

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