<p><strong>174.</strong> Solution set of the equation \(\sqrt{4^x - 2^{x+1} + 1} + \sqrt{4^x - 2^{x+3} + 16} = 3\) is:</p>
Step-by-Step Solution
Key Concept: Recognize that 4^x = (2^x)^2 and use substitution t = 2^x to convert the equation into a recognizable form. Notice that the expressions under the radicals are perfect squares: (2^x - 1)^2 and (2^x - 4)^2.
<p><strong>Step 1:</strong> Let t = 2^x where t > 0. Then 4^x = t^2.</p><p><strong>Step 2:</strong> Rewrite the equation: √(t^2 - 2t + 1) + √(t^2 - 8t + 16) = 3</p><p><strong>Step 3:</strong> Recognize perfect squares: √(t - 1)^2 + √(t - 4)^2 = 3</p><p><strong>Step 4:</strong> This becomes |t - 1| + |t - 4| = 3</p><p><strong>Step 5:</strong> Analyze cases based on critical points t = 1 and t = 4:</p><p> • If 0 < t < 1: (1 - t) + (4 - t) = 3 → 5 - 2t = 3 → t = 1 (boundary, check separately)</p><p> • If 1 ≤ t ≤ 4: (t - 1) + (4 - t) = 3 → 3 = 3 ✓ (always true)</p><p> • If t > 4: (t - 1) + (t - 4) = 3 → 2t - 5 = 3 → t = 4 (boundary, check separately)</p><p><strong>Step 6:</strong> The solution is 1 ≤ t ≤ 4, which means 1 ≤ 2^x ≤ 4</p><p><strong>Step 7:</strong> Taking log: 0 ≤ x ≤ 2</p><p>∴ Answer: D (or x ∈ [0, 2])</p>
Correct Answer: D