Question:

<p><strong>Step 1:</strong> For the first student who got roots 3 and 2 (with constant term wrong, but coefficient of x² correct):</p><p>If roots are 3 and 2, then sum of roots = 5 and product = 6.</p><p>Using Vieta's formulas: if equation is ax² + bx + c = 0, then:</p><p>Sum of roots = -b/a = 5, so b = -5a</p><p>The student copied c incorrectly but a is correct.</p><p><strong>Step 2:</strong> For the second student with roots -6 and 1 (with coefficient of x² wrong):</p><p>Sum of roots = -6 + 1 = -5</p><p>Product of roots = -6 × 1 = -6</p><p>So -b/a = -5, which gives b/a = 5, meaning b = 5a (or the sum of roots = -5)</p><p><strong>Step 3:</strong> From Step 1, the first student's equation has sum of roots = 5, so the coefficient of x in the correct equation gives us: -b/a = -5 (negating to get actual coefficient).</p><p>This means b/a = -5, so the actual sum of roots of correct equation = 5.</p><p><strong>Step 4:</strong> The second student got product of roots = -6 (with a wrong coefficient of x²). The correct product of roots must use the correct a. Since we need to identify which roots work:</p><p>For the correct equation, using the constraint that the first student's a is correct (ratio preserved) and the second student's product constraint:</p><p>If correct roots are 6 and -1:</p><p>Sum = 6 + (-1) = 5 ✓ (matches first student's sum)</p><p>Product = 6 × (-1) = -6 ✓ (matches second student's product)</p><p><strong>Step 5:</strong> Verification: Correct equation is x² - 5x - 6 = 0, which factors as (x - 6)(x + 1) = 0, giving roots 6 and -1.</p><p>∴ Answer: d</p>