2. \( \sqrt{3 x+13}-1=x \)

To solve the equation \( \sqrt{3x+13} - 1 = x \), first, isolate the square root by adding 1 to both sides: \( \sqrt{3x+13} = x + 1 \). Next, square both sides to eliminate the square root: \( 3x + 13 = (x + 1)^2 \). Expanding the right side gives \( 3x + 13 = x^2 + 2x + 1 \). Rearranging it leads to a standard quadratic form: \( x^2 - x - 12 = 0 \). Factoring this results in \( (x - 4)(x + 3) = 0 \), giving solutions \( x = 4 \) and \( x = -3 \). Check both solutions in the original equation to ensure they fit! One of the key takeaways from solving such equations is to always verify your final answers because squaring both sides can introduce extraneous solutions. In this case, checking \( x = 4 \) results in a true statement, while checking \( x = -3 \) results in a contradiction, so the only valid solution is \( x = 4 \).