If 2x - 12x = 5 , x 0, then find the value of x^2 + 116x^2 - 2 = ?

Answer: Option A

$$\begin{array}{rl}& 2x-\frac{1}{2x}=5\\ & \text{Divide by 2 both side}\\ & x-\frac{1}{4x}=\frac{5}{2}\\ & \text{Squaring both side}\\ & \Rightarrow x^2+\frac{1}{16x^2}-2\times x\times \frac{1}{4x}=\frac{25}{4}\\ & \Rightarrow x^2+\frac{1}{16x^2}-\frac{1}{2}=\frac{25}{4}\\ & \Rightarrow x^2+\frac{1}{16x^2}=\frac{25}{4}+\frac{1}{2}\\ & \Rightarrow x^2+\frac{1}{16x^2}=\frac{27}{4}\\ & \text{So, }\\ & x^2+\frac{1}{16x^2}-2\\ & =\frac{27}{4}-2\\ & =\frac{19}{4}\end{array}$$