Question

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 $a c(a-c)+a d(a-d)+b c(b-c)+b d(b-d)$

• A. 0
• B. 8000
• C. 8080
• D. 16000

Answer 1

Answer: (D)

Given $a+b=-20, a b=-2020, c+d=20, c d=2020-09-28$
$a^{2} c-a c^{2}+a^{2} d-a d^{2}+b^{2} c-b^{2} c-b c^{2}+b^{2} d-b d^{2}$
$=\left(a^{2}+b^{2}\right)(c+d)-(a+b)\left(c^{2}+d^{2}\right)$
$=\left((a+b)^{2}-2 a b\right)(c+d)-(a+b)\left((c+d)^{2}-2 c d\right)$
$=(400-2 \times 2020)(20)-(-20)(400-2 \times 2020)$
$=20 \times 800=16000$