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 (

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 $a c (a - c) + a d (a - d) + b c (b - c) + b d (b - d)$ is [2020]

Smart Achievers · Accepted Answer

(4)

Consider the quadratic polynomials in the form of equation

$x^{2} + 20 x - 2020 = 0 \dots (i)$

$x^{2} - 20 x + 2020 = 0 \dots (i i)$

$Since, a and b are roots of the equation (i), then$

$a + b = - 20, a b = - 2020$

$∵$ $c and d are the roots of the equation (i i), then$

$c + d = 20, c d = 2020$

$Now,$

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

$= a^{2} c - a c^{2} + a^{2} d - a d^{2} + b^{2} c - b c^{2} + b^{2} d - b d^{2}$

$= a^{2} (c + d) + b^{2} (c + d) - c^{2} (a + b) - d^{2} (a + b)$

$= (c + d) (a^{2} + b^{2}) - (a + b) (c^{2} + d^{2})$

$= (c + d) ({(a + b)}^{2} - 2 a b) - (a + b) ({(c + d)}^{2} - 2 c d)$

$= 20 [{(20)}^{2} + 4040] + 20 [{(20)}^{2} - 4040]$

$= 20 \times 800 = 16000$

Solution

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 $a c (a - c) + a d (a - d) + b c (b - c) + b d (b - d)$ is [2020]

1 0

2 8000

3 8080

4 16000

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