Question

1. Suppose a, b denote the distinct real roots of the
quadratic polynomial x2 + 20x – 2020 and suppose c,
d denote the distinct complex roots of the quadratic
polynomial x2 – 20x + 2020. Then the value of ac
(a-c) + ad (a - d) + bc (b-c) + bd (b-d) is​

Answer 1

Answer:

(Dx2 + 20x – 2020 = 0 has two roots a,b ∈ R x2 – 20x + 2020 = 0 has two roots c,d ∈ complex ac(a – c) + ad(a – d) + bc(b – c) + bd(b – d) = a2 c – ac2 + a2 d – ad2 + b2 c – bc2 + b2 d – bd2 = a2 (c + d) + b2 (c + d) – c2 (a + b) – d2 (a + b) = (c + d) (a2 + b2) – (a + b) (c2 + d2) = (c + d) ((a + b)2 – 2ab) – (a + b) ((c + d)2 – 2cd) = 20 [(20)2 + 4040] + 20 [(20)2 – 4040] = 20 [(20)2 + 4040 + (20)2 – 4040] = 20 × 800 = 16000