The sum of all real roots of the equation (x^2 - 2x)^2 - 3(x^2 - 2x) - 10 = 0 is:

Step-by-step solution

Let t = x^2 - 2x . The given equation can be rewritten in terms of t as: t^2 - 3t - 10 = 0 Factoring the quadratic in t : (t - 5)(t + 2) = 0 This gives t = 5 or t = -2 . Case 1: t = 5 x^2 - 2x = 5 x^2 - 2x - 5 = 0 The discriminant of this quadratic is D = (-2)^2 - 4(1)(-5) = 4 + 20 = 24 > 0 . Since D > 0 , this equation has two distinct real roots. By Vieta's formulas, the sum of these real roots is - -2 1 = 2 . Case 2: t = -2 x^2 - 2x = -2 x^2 - 2x + 2 = 0 The discriminant of this quadratic is D = (-2)^2 - 4(1)(2)