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Let a and b represent the distinct real roots of the quadratic equation x² + 20x − 2020, and c and d represent the distinct complex roots of the quadratic equation x² − 20x + 2020. What is the value of ac(a − c) + ad(a − d) + bc(b − c) + bd(b − d)?

1. 8000
2. 8080
3. 16000
4. 8080

Correct answer: 16000

Solution

Using a+b=-20, ab=-2020, c+d=20, cd=2020, the expression simplifies to (a^2+b^2)(c+d)-(a+b)(c^2+d^2). With a^2+b^2=400+4040=4440 and c^2+d^2=400-4040=-3640, this is 4440*20-(-20)*(-3640)=88800-72800=16000. Direct substitution of the actual roots confirms 16000.

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