<p>For \(2^{(x-1)(x^2+5x-50)}=1\), the sum of all real values of \(x\) is:</p>
Step-by-Step Solution
Key Concept: An exponential equation with base 2 equals 1 only when the exponent equals 0. So we must solve (x-1)(x²+5x-50) = 0 by finding all roots of both factors.
<p><strong>Step 1:</strong> Since 2^(exponent) = 1, the exponent must equal 0.</p><p>Therefore: (x-1)(x²+5x-50) = 0</p><p><strong>Step 2:</strong> Solve the first factor:</p><p>x - 1 = 0 → x = 1</p><p><strong>Step 3:</strong> Solve the second factor x² + 5x - 50 = 0</p><p>Factoring: (x+10)(x-5) = 0</p><p>→ x = -10 or x = 5</p><p><strong>Step 4:</strong> Verify all three values satisfy the original equation (they do, since the exponent becomes 0).</p><p><strong>Step 5:</strong> Sum all real solutions:</p><p>1 + (-10) + 5 = -4</p><p>∴ Answer: A (Sum = -4)</p>
Correct Answer: A