Use Vieta's formulas instead of solving the quadratic. For x^2 - (17)x + (52) = 0, the roots α and β satisfy α + β = 17 and αβ = 52. 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 = (17)^2 - 2(52) = 289 - (104) = 185. This is why the correct answer is 185.