\( \frac{x_{9}^{2}}{9}+\frac{36}{x^{2}} \), if \( \frac{x}{3}-\frac{6}{x}=1 \)

Given the equation \( \frac{x}{3}-\frac{6}{x}=1 \), we need to solve for \( x \) first. Let's solve the equation \( \frac{x}{3}-\frac{6}{x}=1 \) for \( x \). Solve the equation by following steps: - step0: Solve for \(x\): \(\frac{x}{3}-\frac{6}{x}=1\) - step1: Find the domain: \(\frac{x}{3}-\frac{6}{x}=1,x\neq 0\) - step2: Multiply both sides of the equation by LCD: \(\left(\frac{x}{3}-\frac{6}{x}\right)\times 3x=1\times 3x\) - step3: Simplify the equation: \(x^{2}-18=3x\) - step4: Move the expression to the left side: \(x^{2}-18-3x=0\) - step5: Factor the expression: \(\left(x-6\right)\left(x+3\right)=0\) - step6: Separate into possible cases: \(\begin{align}&x-6=0\\&x+3=0\end{align}\) - step7: Solve the equation: \(\begin{align}&x=6\\&x=-3\end{align}\) - step8: Check if the solution is in the defined range: \(\begin{align}&x=6\\&x=-3\end{align},x\neq 0\) - step9: Find the intersection: \(\begin{align}&x=6\\&x=-3\end{align}\) - step10: Rewrite: \(x_{1}=-3,x_{2}=6\) The solutions to the equation \( \frac{x}{3}-\frac{6}{x}=1 \) are \( x = -3 \) and \( x = 6 \). Now, we need to substitute these values of \( x \) into the expression \( \frac{x_{9}^{2}}{9}+\frac{36}{x^{2}} \) to find the final result. Simplify the expression by following steps: - step0: Solution: \(\frac{x_{9}^{2}}{9}+\frac{36}{x^{2}}\) - step1: Reduce fractions to a common denominator: \(\frac{x_{9}^{2}x^{2}}{9x^{2}}+\frac{36\times 9}{x^{2}\times 9}\) - step2: Reorder the terms: \(\frac{x_{9}^{2}x^{2}}{9x^{2}}+\frac{36\times 9}{9x^{2}}\) - step3: Transform the expression: \(\frac{x_{9}^{2}x^{2}+36\times 9}{9x^{2}}\) - step4: Multiply the numbers: \(\frac{x_{9}^{2}x^{2}+324}{9x^{2}}\) Substitute \( x = -3 \) into the expression \( \frac{x_{9}^{2}}{9}+\frac{36}{x^{2}} \): \( \frac{(-3)^{2}}{9}+\frac{36}{(-3)^{2}} = \frac{9}{9}+\frac{36}{9} = 1+4 = 5 \) Substitute \( x = 6 \) into the expression \( \frac{x_{9}^{2}}{9}+\frac{36}{x^{2}} \): \( \frac{6^{2}}{9}+\frac{36}{6^{2}} = \frac{36}{9}+\frac{36}{36} = 4+1 = 5 \) Therefore, the final result for both \( x = -3 \) and \( x = 6 \) is 5.