Great Quadratic Equation

Algebra Level 5

Suppose that \(x_1\) and \(x_2\) are the positive real solution of \(x^2 - bx + c = 0\) provided that \(x_1^2 + \sqrt{x_2^2 - 2x_2} = 2x_1 - 1\). Find the minimum value of \(b+c\).