Use Vieta's formulas instead of solving the quadratic. For x^2 - (20)x + (99) = 0, the roots α and β satisfy α + β = 20 and αβ = 99. 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 = (20)^2 - 2(99) = 400 - (198) = 202. This is why the correct answer is 202.