Solve the equation. Express numbers in exact simplest form. \[ 8\left(q^{2}-2 q\right)=7 q^{2}-64 \] The solution set is \( \{\square\} \).

We start with the equation \[ 8\left(q^{2} - 2q\right) = 7q^{2} - 64. \] **Step 1. Expand and simplify the equation** First, distribute \(8\) on the left-hand side: \[ 8q^{2} - 16q = 7q^{2} - 64. \] Subtract \(7q^{2}\) and add \(64\) to both sides to set the equation to zero: \[ 8q^{2} - 16q - 7q^{2} + 64 = 0. \] Combine like terms: \[ q^{2} - 16q + 64 = 0. \] **Step 2. Factor the quadratic equation** We observe that the quadratic expression is a perfect square: \[ q^{2} - 16q + 64 = \left(q - 8\right)^{2}. \] Thus, the equation becomes: \[ \left(q - 8\right)^{2} = 0. \] **Step 3. Solve for \(q\)** Since \(\left(q - 8\right)^{2} = 0\), we have: \[ q - 8 = 0. \] Solving for \(q\) gives: \[ q = 8. \] The solution set is \( \{8\} \).