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A quartic equation x⁴ + a x³ + b x² + c x + d = 0 with real coefficients has four non-real roots. One conjugate pair has a sum of 3 + 4i and the remaining conjugate pair has a product of 13 + i. Determine the value of b.

1. 51
2. 49
3. 13
4. 25

Correct answer: 51

Solution

Roots split into pairs (p, q) and their conjugates. The pair with sum 3+4i pairs with conjugate sum 3-4i; product 13+i pairs with 13-i. Using Vieta, b equals the product of the two quadratic constant terms plus the cross terms, yielding 51.

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