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

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8000

8080

16000

answer is D.

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Detailed Solution

Given, a, b denote the distinct real roots of the quadratic polynomial

x^{2} + 20 x - 2020

and c, d denote the distinct complex roots of the quadratic polynomial

x^{2} + 20 x - 2020

.
We need to find the value of

ac (a - c) + ad (a - d) + bc (b - c) + bd (b - d)

. For a quadratic polynomial of the form

a x^{2} + bx + c

,
Sum of roots

= \frac{- coefficiect of x}{coefficient of x^{2}} = \frac{- b}{a}

, and
Product of roots

\frac{constant term}{coefficient of x^{2}} = \frac{c}{a}

Finding sum and product of the roots for the first quadratic polynomial,

x^{2} + 20 x - 2020

a + b = - 20 \dots (1)

and

ab = - 2020

…(2)
Finding sum and product of the roots for the second quadratic polynomial,

x^{2} + 20 x - 2020

c + d = 20

…(3) and

cd = 2020

…(4)
Multiplying and combing terms that make the sum and product of the roots of the given polynomials.
Now, simplifying the expression

ac (a - c) + ad (a - d) + bc (b - c) + bd (b - d),

ac (a - c) + ad (a - d) + bc (b - c) + bd (b - d)

\Rightarrow a^{2} c - a c^{2} + a^{2} d - a d^{2} + b^{2} c - b c^{2} + b^{2} d - b d^{2}

\Rightarrow a^{2} c + a^{2} d + b^{2} c + b^{2} d - a c^{2} - a d^{2} - b d^{2}

{\Rightarrow a}^{2} (c + d) + b^{2} (c + d) - [a (c^{2} + d^{2}) + b (c^{2} + d^{2})]

\Rightarrow (a^{2} + b^{2}) (c + d) - [(c^{2} + d^{2}) (a + b)]

We know that,

{(a + b)}^{2} = a^{2} + b^{2} + 2 ab

\Rightarrow a^{2} + b^{2} = {(a + b)}^{2} - 2 ab

Using the above identity, we get,

\{{(a + b)}^{2} - 2 ab\} (c + d) - [\{{(c + d)}^{2} - 2 cd\} (a + b)]

Substituting the values of

(a + b), ab, (c + d)

and

cd

from (1),(2),(3) and (4), we get:

\{{(- 20)}^{2} - 2 (- 2020)\} (20) - [\{{(20)}^{2} - 2 (2020)\} (- 20)]

\Rightarrow \{400 + 4040\} (20) - [\{400 - 4040\} (- 20)]

\Rightarrow \{4440\} (20) - [\{- 3640\} (- 20)]

\Rightarrow 88800 - 72800

\Rightarrow 16000

Thus,

c (a - c) + ad (a - d) + bc (b - c) + bd (b - d) = 16000

.
Hence, the correct option is 4.

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