The number of real solutions of the equation (x + 5 x + 6)(x - 7 x + 12) = 0 is:

Step-by-step solution

Let t = x . Since x 0 for the square root to be defined, we must have t 0 . Substituting x = t^2 into the given equation, we get: (t^2 + 5t + 6)(t^2 - 7t + 12) = 0 Factoring both quadratic expressions: (t + 2)(t + 3)(t - 3)(t - 4) = 0 This gives the possible values for t as -2, -3, 3, 4 . Since t 0 , we must reject t = -2 and t = -3 . The only valid values are t = 3 and t = 4 . For t = 3 , x = 3 x = 9 . For t = 4 , x = 4 x = 16 . Both x = 9 and x = 16 are valid real solutions. Thus, the number of real solutions is