8. \( x-\sqrt{x-1}=3 \)

To solve the equation \( x - \sqrt{x - 1} = 3 \), you can start by isolating the square root. Rearranging gives \( \sqrt{x - 1} = x - 3 \). Now, square both sides to eliminate the square root: \( x - 1 = (x - 3)^2 \). Expanding the right side gives \( x - 1 = x^2 - 6x + 9 \). Rearranging this into standard form leads to the quadratic equation \( x^2 - 7x + 10 = 0 \). Factoring gives \( (x - 5)(x - 2) = 0 \), so \( x = 5 \) or \( x = 2 \). Finally, check if these solutions satisfy the original equation: both \( x = 5 \) and \( x = 2 \) do! Now you know that solving square root equations often involves isolating and squaring. It's important to check your solutions, as squaring can introduce extraneous solutions that don't satisfy the original equation!