<p>The sum of the non-real roots of \((x^2 + x - 2)(x^2 + x - 3) = 12\) is</p>
Step-by-Step Solution
Key Concept: Substitute y = x² + x to convert the equation into a quadratic in y, then solve for y and extract roots. The non-real roots come from one of the resulting quadratics in x.
<p><strong>Step 1:</strong> Let y = x² + x. The equation becomes:</p><p>(y - 2)(y - 3) = 12</p><p><strong>Step 2:</strong> Expand: y² - 5y + 6 = 12</p><p>y² - 5y - 6 = 0</p><p>(y - 6)(y + 1) = 0</p><p>So y = 6 or y = -1</p><p><strong>Step 3:</strong> Solve for x when y = 6:</p><p>x² + x = 6 → x² + x - 6 = 0 → (x + 3)(x - 2) = 0</p><p>x = -3 or x = 2 (both real)</p><p><strong>Step 4:</strong> Solve for x when y = -1:</p><p>x² + x = -1 → x² + x + 1 = 0</p><p>Discriminant = 1 - 4 = -3 < 0 (non-real roots)</p><p><strong>Step 5:</strong> By Vieta's formulas for x² + x + 1 = 0:</p><p>Sum of non-real roots = -1/1 = -1</p><p>∴ Answer: A</p>
Correct Answer: A