JEE Advanced · Mathematics · 2. Quadratic Equations

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 $16000$

Verified Solution

Answer & Solution

Correct Answer

(D) $16000$

Step-by-step Solution

Detailed explanation

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})

$=\left\{(a+b)^2-2 a b\right\}(c+d)-(a+b)$$\left\{(c+d)^2-2 c d\right\} $
$ \left.=\{(-20)^2-2(-2020)\right\}(20)-(-20)$$\left\{(20)^2-2(2020)\right\}$ (using equations (i) & (ii)

= 16000

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Suppose a, b denote the distinct real roots of the quadratic polynomial x 2 +20x−2020 and suppose c, d denote the distinct complex roots of the quadratic polynomial x2-20x+2020. Then the value of aca−c+ada−d+bcb−c+bdb−d

Step-by-step Solution

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