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

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

TestHub · Accepted Answer

Finding the sum and product of roots of the two given quadratic equations i.e.

$a + b = - 20 & a b = - 2020 . . . . . . . . . . . (i)$

$c + d = 20 & c d = 2020 . . . . . . . . . . . (i i)$

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

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

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

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

$= \{{(- 20)}^{2} - 2 (- 2020)\} (20) - (- 20) \{{(20)}^{2} - 2 (2020)\}$ (using equations $(i) & (i i)$ )

$= 16000$

Mathematics - Quadratic Equation Question with Solution | TestHub

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