<p>The number of real roots of the equation \(5 + |2^x - 1| = 2^x(2^x - 2)\) is __________.</p>
Step-by-Step Solution
Key Concept: Substitute y = 2^x to convert into an absolute value equation in y, then systematically handle cases where (2^x - 1) is positive or negative. The key is recognizing that y = 2^x > 0 always, which constrains valid solutions.
<p><strong>Step 1:</strong> Substitute y = 2^x where y > 0. The equation becomes: 5 + |y - 1| = y(y - 2) = y² - 2y</p><p><strong>Step 2:</strong> <strong>Case 1:</strong> If y ≥ 1, then |y - 1| = y - 1<br/>5 + (y - 1) = y² - 2y<br/>4 + y = y² - 2y<br/>y² - 3y - 4 = 0<br/>(y - 4)(y + 1) = 0<br/>y = 4 or y = -1<br/>Since y > 0 and y ≥ 1, only y = 4 is valid. This gives 2^x = 4, so x = 2.</p><p><strong>Step 3:</strong> <strong>Case 2:</strong> If 0 < y < 1, then |y - 1| = 1 - y<br/>5 + (1 - y) = y² - 2y<br/>6 - y = y² - 2y<br/>y² - y - 6 = 0<br/>(y - 3)(y + 2) = 0<br/>y = 3 or y = -2<br/>Since we need 0 < y < 1, neither solution is valid in this range.</p><p><strong>Step 4:</strong> Verification: For x = 2: LHS = 5 + |4 - 1| = 5 + 3 = 8; RHS = 4(4 - 2) = 4(2) = 8 ✓</p><p>∴ <strong>Answer: 1</strong></p>
Correct Answer: 1