Solve the equation by the zero-factor property. \( a^{2}-12 a+32=0 \)

To solve the quadratic equation \( a^{2} - 12a + 32 = 0 \) using the **zero-factor property**, follow these steps: 1. **Factor the quadratic equation:** We need to express the quadratic in the form \((a - r)(a - s) = 0\), where \( r \) and \( s \) are numbers that satisfy: - \( r \times s = 32 \) (the constant term) - \( r + s = 12 \) (the coefficient of the middle term, with a sign change) Let's find the suitable numbers: - **Possible pairs that multiply to 32:** - \( 1 \) and \( 32 \) (sum = 33) - \( 2 \) and \( 16 \) (sum = 18) - \( 4 \) and \( 8 \) (sum = 12) ✅ The pair \( 4 \) and \( 8 \) adds up to \( 12 \). Therefore, the factors are: \[ (a - 4)(a - 8) = 0 \] 2. **Apply the Zero-Product Property:** If \((a - 4)(a - 8) = 0\), then either: \[ a - 4 = 0 \quad \text{or} \quad a - 8 = 0 \] 3. **Solve for \( a \):** - \( a - 4 = 0 \) ⟹ \( a = 4 \) - \( a - 8 = 0 \) ⟹ \( a = 8 \) **Solution:** The solutions are \( a = 4 \) and \( a = 8 \).