Let b and c be positive integers. If the equation |x^2 - bx + c| - 2x + 7 = 0 has x = 5 and x = 2 + 3 as two of its roots, then the value of b + c is

Step-by-step solution

Given the equation |x^2 - bx + c| - 2x + 7 = 0 . Since x = 2 + 3 is a root, we substitute it into the equation: |(2 + 3 )^2 - b(2 + 3 ) + c| - 2(2 + 3 ) + 7 = 0 |7 + 4 3 - 2b - b 3 + c| = 4 + 2 3 - 7 |(7 - 2b + c) + (4 - b) 3 | = -3 + 2 3 Since the absolute value of a real number is non-negative, and -3 + 2 3 = 12 - 9 > 0 , the equation is valid. Removing the absolute value gives two cases: Case 1: (7 - 2b + c) + (4 - b) 3 = -3 + 2 3 Comparing rational and irrational parts (since b and c are integers): 4 - b = 2 b