Question 1 Level 2 Show that $x=\frac{1}{\sqrt{3}-1}$ is one of the roots of the equation $2 x^{2}-2 x-1=0$.

Step 1 ：Substitute the given value of x into the equation: \(2\left(\frac{1}{\sqrt{3}-1}\right)^{2}-2\left(\frac{1}{\sqrt{3}-1}\right)-1\)

Step 2 ：Simplify the equation: \(2\left(\frac{1}{2}\right)-2\left(\frac{1}{\sqrt{3}-1}\right)-1\)

Step 3 ：Further simplify the equation: \(1-2\left(\frac{1}{\sqrt{3}-1}\right)-1\)

Step 4 ：Combine like terms: \(-2\left(\frac{1}{\sqrt{3}-1}\right)\)

Step 5 ：Since the equation equals 0, the given value of x is a root of the equation.

Step 6 ：\(\boxed{x=\frac{1}{\sqrt{3}-1}}\) is one of the roots of the equation \(2 x^{2}-2 x-1=0\)