If the equation 9^x - (a+3)3^x + (a^2 - 4) = 0 has exactly one positive real root and exactly one negative real root, then the set of all possible values of a is :

Step-by-step solution

Let t = 3^x . The given equation becomes a quadratic in t : f(t) = t^2 - (a+3)t + (a^2 - 4) = 0 We are given that the original equation has one negative real root and one positive real root. If x If x > 0 , then 3^x = t (1, ) . Thus, the quadratic equation f(t) = 0 must have one root in the interval (0, 1) and the other root in the interval (1, ) . This implies that the roots t₁ and t₂ satisfy 0 For the roots of f(t) to lie on either side of 1 , we must have f(1) f(1) = 1 - (a+3) + a^2 - 4 a^2 - a - 6 (a-3)(a+2) Ad