Solve the given equation by the zero-factor property. \[ x^{2}-64=0 \]

To solve the equation \( x^{2} - 64 = 0 \) using the **zero-factor property**, follow these steps: ### Step 1: Factor the Equation The given equation is a **difference of squares**, which can be factored using the formula: \[ a^{2} - b^{2} = (a - b)(a + b) \] Apply this to the equation: \[ x^{2} - 64 = x^{2} - 8^{2} = (x - 8)(x + 8) \] ### Step 2: Apply the Zero-Factor Property The **zero-factor property** states that if a product of two factors is zero, then at least one of the factors must be zero. Therefore: \[ (x - 8)(x + 8) = 0 \] This gives two possible equations: 1. \( x - 8 = 0 \) 2. \( x + 8 = 0 \) ### Step 3: Solve for \( x \) 1. **First Equation:** \[ x - 8 = 0 \] \[ x = 8 \] 2. **Second Equation:** \[ x + 8 = 0 \] \[ x = -8 \] ### Conclusion The solutions to the equation \( x^{2} - 64 = 0 \) are: \[ x = 8 \quad \text{and} \quad x = -8 \] **Verification:** Plugging these values back into the original equation: - For \( x = 8 \): \[ (8)^2 - 64 = 64 - 64 = 0 \] - For \( x = -8 \): \[ (-8)^2 - 64 = 64 - 64 = 0 \] Both solutions satisfy the equation. **Final Answer:** The solutions are \( x = 8 \) and \( x = -8 \).