Use Vieta's formulas instead of solving the quadratic. For x^2 - (11)x + (24) = 0, the roots α and β satisfy α + β = 11 and αβ = 24. 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 = (11)^2 - 2(24) = 121 - (48) = 73. This is why the correct answer is 73.