Suppose a, b denote the distinct real roots of the quadratic polynomial x2+20 x-2020 and suppose c, d denote the distinct complex roots of the quadratic polynomial x2-20 x+2020. Then the value of a c(a-c)+a d(a-d)+b c(b-c)+b d(b-d)

Question

Suppose a, b denote the distinct real roots of the quadratic polynomial x2+20 x-2020 and suppose c, d denote the distinct complex roots of the quadratic polynomial x2-20 x+2020. Then the value of a c(a-c)+a d(a-d)+b c(b-c)+b d(b-d) (A) 0 (B) 8000 (C) 8080 (D) 16000

Tardigrade Admin · Accepted Answer

Given a+b=-20, a b=-2020, c+d=20, c d=2020-09-28 a2 c-a c2+a2 d-a d2+b2 c-b2 c-b c2+b2 d-b d2 =(a2+b2)(c+d)-(a+b)(c2+d2) =((a+b)2-2 a b)(c+d)-(a+b)((c+d)2-2 c d) =(400-2 × 2020)(20)-(-20)(400-2 × 2020) =20 × 800=16000

Q. 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)

0

8000

8080

16000

Given a+b=−20,ab=−2020,c+d=20,cd=2020−09−28 a2c−ac2+a2d−ad2+b2c−b2c−bc2+b2d−bd2 =(a2+b2)(c+d)−(a+b)(c2+d2) =((a+b)2−2ab)(c+d)−(a+b)((c+d)2−2cd) =(400−2×2020)(20)−(−20)(400−2×2020) =20×800=16000

Q. Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^{2} + 20 x - 2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^{2} - 20 x + 2020$ . Then the value of $a c (a - c) + a d (a - d) + b c (b - c) + b d (b - d)$