If x + 9/x = 6 , then ( x^2 + 9x^2 ) is equal to?

Answer: Option C

$$\begin{array}{rl}& x+\frac{9}{x}=6\\ & \text{Take value of x}\\ & \text{Let }x=3\\ & 3+\frac{9}{3}=6\left(\text{Proved}\right)\\ & \text{So, }x=3\\ & \therefore x^2+\frac{9}{x^2}\\ & =9+\frac{9}{9}\\ & =10\\ & \\ & \mathbf{Alternate}:\\ & x+\frac{9}{x}=6\\ & \text{On squaring,}\\ & \Rightarrow {\left(x+\frac{9}{x}\right)}^2=36\\ & \Rightarrow x^2+\frac{81}{x^2}+2\times x\times \frac{9}{x}=36\\ & \Rightarrow x^2+\frac{81}{x^2}-18=0\\ & \Rightarrow {\left(x-\frac{9}{x}\right)}^2=0\\ & \Rightarrow x=\frac{9}{x}\\ & \Rightarrow x^2=9\\ & \text{Hence, }\left(x^2+\frac{9}{x^2}\right)\\ & =\left(9+\frac{9}{9}\right)\\ & =10\end{array}$$