Algebra + Geometry

The equation $C(x)$ can be represented as $(2x-3)(2x-3)(x-q)$ . Since the roots are the sides of $ABC$ and $AB=AC$ , $(2x-3) = AB = AC$ . As a result, the congruent sides both have a length of $2x-3 = 0$ , so $x=1.5$ . Solving the exponential equation yields $q<-1.666...$ and $q>2$ . Since $q$ is a side of the triangle, as it is a root in the equation, $q<-1.6666...$ and be eliminated as triangle sides cannot be negative. By Triangle Inequality, $1.5+1.5>q$ , so $q<3$ . From the exponential equation, we know $q>2$ , so $2<q<3$ . Since we know that two of the sides are both $1.5$ , we can use the law of cosines to find the minimum and the maximum for $cos(A)$ using the inequalities $2(1.5^2)(1-cos(A))>2^2$ and $2(1.5^2)(1-cos(A))<3^2$ . This yields $-0.1111... < cos(A) < 1$ , so by substitution $1 - 0.111... = \frac{8}{9}$ , so $8*9=72$ .