1. **State the problem:** Solve the equation $$\frac{(x-3)(x+1)}{\sqrt{\ln(x-1)}} = 0$$ and find the arithmetic average of its roots. 2. **Understand the equation:** A fraction equals zero if and only if its numerator is zero and the denominator is defined (not zero or undefined). 3. **Set numerator equal to zero:** $$ (x-3)(x+1) = 0 $$ This gives roots: $$ x = 3 \quad \text{or} \quad x = -1 $$ 4. **Check domain restrictions from denominator:** The denominator is $$ \sqrt{\ln(x-1)} $$, so: - The argument of the logarithm must be positive: $$ x-1 > 0 \Rightarrow x > 1 $$ - The logarithm must be non-negative for the square root to be real: $$ \ln(x-1) \geq 0 \Rightarrow x-1 \geq 1 \Rightarrow x \geq 2 $$ 5. **Check which roots satisfy domain:** - For $$ x=3 $$: $$ 3 \geq 2 $$ valid. - For $$ x=-1 $$: $$ -1 > 1 $$ false, so discard. 6. **Only root is $$ x=3 $$** 7. **Arithmetic average of roots:** Since only one root is valid, the average is: $$ \frac{3}{1} = 3 $$ **Final answer:** $$3$$