1. **State the problem:** A theatre has a central block with $n$ seats per row. On each side of this block, there are blocks with 4 seats per row. The number of rows is 5 less than the total number of seats per row. The total number of seats in the theatre is 126. We need to find the value of $n$. 2. **Define variables and expressions:** - Central block seats per row: $n$ - Side blocks seats per row: 4 each side, so total side seats per row = $4 + 4 = 8$ - Total seats per row = central block + side blocks = $n + 8$ - Number of rows = total seats per row $- 5 = (n + 8) - 5 = n + 3$ 3. **Write the equation for total seats:** Total seats = (seats per row) $\times$ (number of rows) $$ (n + 8)(n + 3) = 126 $$ 4. **Expand and simplify:** $$ (n + 8)(n + 3) = n^2 + 3n + 8n + 24 = n^2 + 11n + 24 $$ So, $$ n^2 + 11n + 24 = 126 $$ 5. **Bring all terms to one side:** $$ n^2 + 11n + 24 - 126 = 0 \\ n^2 + 11n - 102 = 0 $$ 6. **Solve the quadratic equation:** Use the quadratic formula: $$ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \\ a=1, b=11, c=-102 $$ Calculate discriminant: $$ \Delta = 11^2 - 4 \times 1 \times (-102) = 121 + 408 = 529 $$ Square root: $$ \sqrt{529} = 23 $$ So, $$ n = \frac{-11 \pm 23}{2} $$ 7. **Find the two possible values:** - $n = \frac{-11 + 23}{2} = \frac{12}{2} = 6$ - $n = \frac{-11 - 23}{2} = \frac{-34}{2} = -17$ Since $n$ represents number of seats per row, it must be positive. So, $$ \boxed{n = 6} $$ **Final answer:** The central block has 6 seats per row.