4.2 Quadratic equations and inequations — ગણિત ધોરણ 11 સાયન્સ — Question

.==> $\frac{3}{4}{t^2} + t - \frac{5}{4} = \frac{1}{2}\left( {\frac{1}{t}} \right)$
By putting $t = {\log _2}x$ so that ${\log _x}2 = \frac{1}{t}$ because
${\log _2}x{\log _x}2 = 1$.

==> $3{t^3} + 4{t^2} - 5t - 2 = 0\,\,\,\, \Rightarrow (t - 1)(t + 2)(3t + 1) = 0$

==> ${\log _2}x = t = 1, - 2, - \frac{1}{3}$

==> $x = 2,{2^{ - 2}},{2^{ - 1/3}}$or $x = 2,\frac{1}{4},\frac{1}{{{2^{1/3}}}}$

Thus the given equation has exactly three real solutions out of which exactly one is irrational namely $\frac{1}{{{2^{1/3}}}}$.