Consider the equation x^2 - 6x - 5|x - 3| + k = 0 , where k is a real parameter. The sum of all positive integral values of k for which the equation has exactly two distinct real roots is

Step-by-step solution

Rewrite the given equation by completing the square for the non-modulus terms: x^2 - 6x + 9 - 9 - 5|x - 3| + k = 0 (x - 3)^2 - 5|x - 3| + k - 9 = 0 Let u = |x - 3| . Note that u 0 . The equation becomes a quadratic in u : u^2 - 5u + (k - 9) = 0 For each positive real root u , the equation |x - 3| = u yields exactly two distinct real roots for x ( x = 3 u ). If u = 0 , it yields exactly one real root ( x = 3 ). If u To have exactly two distinct real roots for x , the quadratic in u must have exactly one positive roo