If \( x^{2}-7 x+1=0 \), what is \( x+\frac{1}{x}-7 ? \)

Given the equation \( x^{2}-7x+1=0 \), we need to find the value of \( x+\frac{1}{x}-7 \). To find the value of \( x+\frac{1}{x}-7 \), we first need to solve the equation \( x^{2}-7x+1=0 \) to find the values of \( x \). Let's solve the equation \( x^{2}-7x+1=0 \) using the quadratic formula. Solve the quadratic equation by following steps: - step0: Solve using the quadratic formula: \(x^{2}-7x+1=0\) - step1: Solve using the quadratic formula: \(x=\frac{7\pm \sqrt{\left(-7\right)^{2}-4}}{2}\) - step2: Simplify the expression: \(x=\frac{7\pm \sqrt{45}}{2}\) - step3: Simplify the expression: \(x=\frac{7\pm 3\sqrt{5}}{2}\) - step4: Separate into possible cases: \(\begin{align}&x=\frac{7+3\sqrt{5}}{2}\\&x=\frac{7-3\sqrt{5}}{2}\end{align}\) - step5: Rewrite: \(x_{1}=\frac{7-3\sqrt{5}}{2},x_{2}=\frac{7+3\sqrt{5}}{2}\) The solutions to the equation \( x^{2}-7x+1=0 \) are \( x_{1}=\frac{7-3\sqrt{5}}{2} \) and \( x_{2}=\frac{7+3\sqrt{5}}{2} \). Now, we can substitute these values of \( x \) into the expression \( x+\frac{1}{x}-7 \) to find the value. Let's calculate it. Calculate the value by following steps: - step0: Calculate: \(\frac{7-3\sqrt{5}}{2}+\frac{2}{7-3\sqrt{5}}-7\) - step1: Reduce fractions to a common denominator: \(\frac{\left(7-3\sqrt{5}\right)\left(7-3\sqrt{5}\right)}{2\left(7-3\sqrt{5}\right)}+\frac{2\times 2}{\left(7-3\sqrt{5}\right)\times 2}-\frac{7\left(14-6\sqrt{5}\right)}{14-6\sqrt{5}}\) - step2: Calculate: \(\frac{94-42\sqrt{5}}{14-6\sqrt{5}}+\frac{4}{14-6\sqrt{5}}-\frac{98-42\sqrt{5}}{14-6\sqrt{5}}\) - step3: Transform the expression: \(\frac{94-42\sqrt{5}+4-\left(98-42\sqrt{5}\right)}{14-6\sqrt{5}}\) - step4: Remove the parentheses: \(\frac{94-42\sqrt{5}+4-98+42\sqrt{5}}{14-6\sqrt{5}}\) - step5: Calculate: \(\frac{0}{14-6\sqrt{5}}\) - step6: Calculate: \(0\) The value of \( x+\frac{1}{x}-7 \) is 0 when \( x=\frac{7-3\sqrt{5}}{2} \). Similarly, when \( x=\frac{7+3\sqrt{5}}{2} \), the value of \( x+\frac{1}{x}-7 \) is also 0. Therefore, the value of \( x+\frac{1}{x}-7 \) is 0 for both solutions of the equation \( x^{2}-7x+1=0 \).