If the equation 2 ^2(x) + (2a - 2 ) (x) - a 2 = 0 has exactly 5 distinct solutions in the interval x [0, 3 ] , then the value of the parameter a is:

Step-by-step solution

The given equation is 2 ^2(x) + 2a (x) - 2 (x) - a 2 = 0 . Factorizing by grouping the terms: 2 (x)( (x) + a) - 2 ( (x) + a) = 0 (2 (x) - 2 )( (x) + a) = 0 This gives (x) = 1 2 or (x) = -a . Let us first count the solutions for (x) = 1 2 in the interval [0, 3 ] : In [0, ] , x = 4 , 3 4 (2 solutions). In [ , 2 ] , there are no solutions. In [2 , 3 ] , x = 9 4 , 11 4 (2 solutions). Thus, (x) = 1 2 contributes exactly 4 distinct solutions. Since the equation must have exactly 5 distinct solutions in total, the second