Key Concepts and Formulas

• Exponential Property: $b^y = 1$ if and only if $y = 0$, provided $b \neq 0, 1, -1$.
• Factoring Quadratic Expressions: Finding two numbers whose product equals the constant term and whose sum equals the coefficient of the linear term.
• Zero Product Property: If $ab = 0$, then $a = 0$ or $b = 0$ (or both).

Step-by-Step Solution

Step 1: Understanding the Exponential Equation

• What and Why: We recognize the equation as an exponential equation where the base is 2 and the exponent is an expression involving $x$. Our goal is to find the values of $x$ that satisfy the equation.
• Equation: $2^{(x-1)(x^2 + 5x - 50)} = 1$

Step 2: Applying the Exponential Property

• What and Why: Since the base is 2 (which is not 0, 1, or -1), we can use the exponential property. The only way for $2$ raised to some power to equal $1$ is if that power is $0$.
• Application: Setting the exponent equal to zero: $(x-1)(x^2 + 5x - 50) = 0$

Step 3: Factoring the Quadratic Expression

• What and Why: We need to factor the quadratic expression $x^2 + 5x - 50$ to find its roots. This will help us solve the entire polynomial equation.
• Finding Factors: We look for two numbers that multiply to $-50$ and add to $5$. These numbers are $10$ and $-5$.
• Factorization: $x^2 + 5x - 50 = (x + 10)(x - 5)$

Step 4: Completing the Factorization of the Cubic Equation

• What and Why: Substitute the factored quadratic expression back into the equation to get the complete factorization of the cubic polynomial.
• Substitution: $(x-1)(x+10)(x-5) = 0$

Step 5: Finding the Real Roots

• What and Why: Apply the Zero Product Property. Set each factor equal to zero and solve for $x$.
• Solutions:
  • $x - 1 = 0 \implies x = 1$
  • $x + 10 = 0 \implies x = -10$
  • $x - 5 = 0 \implies x = 5$

Step 6: Verifying the Roots

• What and Why: Check that the roots obtained are indeed real numbers.
• Verification: The roots $1$, $-10$, and $5$ are all real numbers.

Step 7: Calculating the Sum of the Real Roots

• What and Why: Add the real roots together to find the sum.
• Calculation: $1 + (-10) + 5 = 1 - 10 + 5 = -9 + 5 = -4$

Common Mistakes & Tips

• Mistake: Forgetting the conditions for applying the exponential property ($b \neq 0, 1, -1$). In this problem, the base is 2, so the property holds.
• Mistake: Making errors in factoring the quadratic expression. Double-check your factors.
• Tip: Always verify that the solutions you obtain are real numbers, as specified in the problem statement.

Summary

We solved the exponential equation by setting the exponent equal to zero and then factoring the resulting polynomial equation. The real roots were found to be $1$, $-10$, and $5$. The sum of these real roots is $-4$.

Final Answer

The final answer is \boxed{-4}, which corresponds to option (C).