Use Vieta's formulas instead of solving the quadratic. For x^2 - (0)x + (-64) = 0, the roots α and β satisfy α + β = 0 and αβ = -64. The expression we need is α^2 + β^2. Use the identity α^2 + β^2 = (α + β)^2 - 2αβ. So α^2 + β^2 = (0)^2 - 2(-64) = 0 - (-128) = 128. This is why the correct answer is 128.