Find the value of x, if (2x + 3) : (5x - 38) is the duplicate ratio of $\sqrt{5} : \sqrt{6}$

According to question,

$\Rightarrow \dfrac{2x + 3}{5x - 38} = \dfrac{(\sqrt{5})^2}{(\sqrt{6})^2} \\[1em] \Rightarrow \dfrac{2x + 3}{5x - 38} = \dfrac{5}{6} \\[1em] \Rightarrow 6(2x + 3) = 5(5x - 38) \\[1em] \Rightarrow 12x + 18 = 25x - 190 \\[1em] \Rightarrow 25x - 12x = 190 + 18 \\[1em] \Rightarrow 13x = 208 \\[1em] \Rightarrow x = 16.$

Hence, x = 16.