In algebra, solving a quadratic equation involves transitioning through several specific stages. Match each stage of the solution for $(k + 1)(k - 1) = 8$ with its correct algebraic representation.

In a professional technical workshop, a junior technician is verifying a safety clearance calculation where the variable $k$ must satisfy the equation $(k + 1)(k - 1) = 8$. Arrange the following algebraic steps in the correct logical sequence to find the valid values for $k$.

A facility manager is using the equation $(k + 1)(k - 1) = 8$ to calculate the required clearance for a new equipment installation. According to the standard procedure for solving this equation, which of the following represents the equation once it has been rewritten in standard form ($ak^2 + bk + c = 0$)?

A quality control technician is calculating the tolerance factor $k$ for a mechanical component using the equation $(k + 1)(k - 1) = 8$. After rewriting the equation in the standard form $k^2 - 9 = 0$ and factoring it as $(k - 3)(k + 3) = 0$, the technician determines the two possible values for $k$. One value is 3; the other value is ____.

Identifying Algebraic Rules in Quality Control

A manufacturing technician is using the equation $(k + 1)(k - 1) = 8$ to determine calibration settings for a machine. Because the left side is already factored, the correct first step is to set each factor equal to 8, yielding $k + 1 = 8$ and $k - 1 = 8$.