Suppose a , b denote the distinct real roots of the quadrtic polynomial `x^(2)+20x-2020` and suppose c,d denote the distinct complex roots of the quadratic polynomial `x^(2)-20x+2020` , then the value of `ac(a-c)+ad(a-d) bc(b-c)bd(b-d)` is

Options

A. 0
B. 8000
C. 8080
D. 16000

Answer

To solve the problem, we need to find the value of the expression \( ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d) \) given the roots of two quadratic polynomials.

### Step 1: Find the roots of the first polynomial \( x^2 + 20x - 2020 = 0 \)

Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1, b = 20, c = -2020 \).

Calculating the discriminant:
\[ D = b^2 - 4ac = 20^2 - 4 \cdot 1 \cdot (-2020) = 400 + 8080 = 8480 \]

Now, substituting into the quadratic formula:
\[ x = \frac{-20 \pm \sqrt{8480}}{2} \]
Calculating \( \sqrt{8480} \):
\[ \sqrt{8480} = \sqrt{16 \cdot 530} = 4\sqrt{530} \]
Thus, the roots \( a \) and \( b \) are:
\[ a, b = \frac{-20 \pm 4\sqrt{530}}{2} = -10 \pm 2\sqrt{530} \]

### Step 2: Find the roots of the second polynomial \( x^2 - 20x + 2020 = 0 \)

Using the quadratic formula again:
\[ x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4 \cdot 1 \cdot 2020}}{2 \cdot 1} \]
Calculating the discriminant:
\[ D = (-20)^2 - 4 \cdot 1 \cdot 2020 = 400 - 8080 = -7680 \]
Since the discriminant is negative, the roots \( c \) and \( d \) are complex:
\[ c, d = \frac{20 \pm \sqrt{-7680}}{2} = 10 \pm i\sqrt{7680} \]
Calculating \( \sqrt{7680} \):
\[ \sqrt{7680} = \sqrt{256 \cdot 30} = 16\sqrt{30} \]
Thus, the roots \( c \) and \( d \) are:
\[ c, d = 10 \pm 16i\sqrt{30} \]

### Step 3: Calculate the expression \( ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d) \)

First, we can simplify the expression:
\[ ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d) = a^2c + a^2d - ac(c+d) + b^2c + b^2d - bc(c+d) \]
Combining like terms:
\[ = a^2(c+d) + b^2(c+d) - (ac + bc)(c+d) \]
Factoring out \( (c+d) \):
\[ = (a^2 + b^2 - (a+b)(c+d))(c+d) \]

### Step 4: Calculate \( a+b \), \( ab \), \( c+d \), and \( cd \)

From the first polynomial:
- \( a + b = -20 \)
- \( ab = -2020 \)

From the second polynomial:
- \( c + d = 20 \)
- \( cd = 2020 \)

### Step 5: Substitute values into the expression

Substituting \( a+b \) and \( c+d \):
\[ = (a^2 + b^2 - (-20)(20))(20) \]
Calculating \( a^2 + b^2 \):
\[ a^2 + b^2 = (a+b)^2 - 2ab = (-20)^2 - 2(-2020) = 400 + 4040 = 4440 \]
Thus, substituting back:
\[ = (4440 + 400)(20) = 4840 \cdot 20 = 96800 \]

### Final Answer
The value of \( ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d) \) is \( \boxed{96800} \).