<p>The number of real solutions of \(|x - 2\sqrt{5 - 4x - x^2}| = 16\) is/are</p>
Step-by-Step Solution
Key Concept: Recognize that the expression under the square root must be non-negative, which constrains x to a specific interval. Then substitute u = √(5 - 4x - x²) to convert the absolute value equation into a manageable form with domain restrictions.
<p><strong>Step 1:</strong> Find the domain. For √(5 - 4x - x²) to be real: 5 - 4x - x² ≥ 0 ⟹ -x² - 4x + 5 ≥ 0 ⟹ x² + 4x - 5 ≤ 0 ⟹ (x+5)(x-1) ≤ 0. Thus <strong>-5 ≤ x ≤ 1</strong>.</p><p><strong>Step 2:</strong> Let u = √(5 - 4x - x²) where u ≥ 0. The equation becomes |x - 2u| = 16.</p><p><strong>Step 3:</strong> Case 1: x - 2u = 16 ⟹ x = 16 + 2u. Since -5 ≤ x ≤ 1, we need -5 ≤ 16 + 2u ≤ 1 ⟹ -21 ≤ 2u ≤ -15. But u ≥ 0, so this case gives <strong>no solutions</strong>.</p><p><strong>Step 4:</strong> Case 2: x - 2u = -16 ⟹ x = 2u - 16. Since -5 ≤ x ≤ 1, we need -5 ≤ 2u - 16 ≤ 1 ⟹ 11 ≤ 2u ≤ 17 ⟹ 5.5 ≤ u ≤ 8.5.</p><p><strong>Step 5:</strong> Substitute x = 2u - 16 into u² = 5 - 4x - x²: u² = 5 - 4(2u - 16) - (2u - 16)² = 5 - 8u + 64 - (4u² - 64u + 256) = 69 - 8u - 4u² + 64u - 256 = -4u² + 56u - 187.</p><p><strong>Step 6:</strong> This gives 5u² - 56u + 187 = 0. Using the discriminant: Δ = 56² - 4(5)(187) = 3136 - 3740 = -604 < 0. <strong>No real solutions</strong> from this quadratic.</p><p><strong>Step 7:</strong> Since neither case yields solutions within the domain constraints, the number of real solutions is <strong>0</strong>.</p><p>∴ Answer: C (0)</p>
Correct Answer: C