<p>Number of integral roots of \(|x-1||x^2 - 2| = 2\) is</p>
Step-by-Step Solution
Key Concept: Split the equation into cases based on the sign of (x-1) and (x²-2), then solve each case separately. The absolute value equation converts to a piecewise problem where you must check which solutions satisfy their respective case conditions.
<p><strong>Step 1:</strong> Identify critical points where expressions change sign: x = 1, x = ±√2 ≈ ±1.414</p><p><strong>Step 2:</strong> Create cases based on intervals: (-∞,-√2), (-√2,1), (1,√2), (√2,∞)</p><p><strong>Case 1 (x < -√2):</strong> (1-x)(2-x²) = 2 → x² - x - 1 = 0 → x = (1-√5)/2 ≈ -0.618 (rejected, doesn't satisfy x < -√2)</p><p><strong>Case 2 (-√2 ≤ x < 1):</strong> (1-x)(2-x²) = 2 → x² - x - 1 = 0 → x = (1-√5)/2 ≈ -0.618 ✓ (satisfies interval)</p><p><strong>Case 3 (1 ≤ x < √2):</strong> (x-1)(2-x²) = 2 → -x³ + x² + 2x - 4 = 0 → Testing: x = √2 gives 0 ≠ 2; no integer solutions in [1, 1.414)</p><p><strong>Case 4 (x ≥ √2):</strong> (x-1)(x²-2) = 2 → x³ - x² - 2x = 0 → x(x² - x - 2) = 0 → x(x-2)(x+1) = 0 → x = 2 ✓ (satisfies x ≥ √2)</p><p><strong>Step 3:</strong> Check x = 0: |0-1||0-2| = 1·2 = 2 ✓</p><p><strong>Step 4:</strong> Verify integer solutions: x = 0 and x = 2 are the only integers satisfying the equation</p><p>∴ <strong>Number of integral roots = 2</strong> (Answer: D)</p>
Correct Answer: D