If 4x = √5 + 2, then the value of ( x - 116x ) is

Answer: Option A

$$\begin{array}{rl}& \text{Given,}4x=\sqrt{5}+2\\ & \Rightarrow 16x=4\left(\sqrt{5}+2\right)\\ & \Rightarrow 16x=4\sqrt{5}+8\\ & \therefore \frac{1}{16x}=\frac{1}{4\sqrt{5}+8}\\ & \Rightarrow \frac{1}{16x}=\frac{4\sqrt{5}-8}{\left(4\sqrt{5}+8\right)\left(4\sqrt{5}-8\right)}\\ & \left[\text{Rationalising the denominator}\right]\\ & \Rightarrow \frac{1}{16x}=\frac{4\sqrt{5}-8}{80-64}\\ & \Rightarrow \frac{1}{16x}=\frac{4\sqrt{5}-8}{16}\\ & \Rightarrow \frac{1}{16x}=\frac{4\left(\sqrt{5}-2\right)}{16}\\ & \Rightarrow \frac{1}{16x}=\frac{\sqrt{5}-2}{4}\\ & \therefore \left(x-\frac{1}{16x}\right)\\ & =\frac{\sqrt{5}+2}{4}-\frac{\sqrt{5}-2}{4}\\ & =\frac{\sqrt{5}+2-\sqrt{5}+2}{4}\\ & =\frac{4}{4}\\ & =1\end{array}$$