Suppose a, b denote the distinct real roots of the quadratic polynomial x² + 20x – 2020 and suppose

Q: Why can't we just use the quadratic formula to find $a, b, c, d$?

That is the classic trap! If you try to find the roots, you will get messy expressions involving square roots of $2020$ and complex numbers. The JEE Advanced examiners design these problems specifically to reward students who spot the algebraic structure rather than those who brute-force the calculation. Always look for symmetry first!

Q: I got lost in the expansion. Is there a trick to grouping?

The trick is to look for common factors. When you expand the expression, you get eight terms. By grouping terms that share common variables like $a^2$ or $c^2$, you force the expression to reveal the sums $(a+b)$ and $(c+d)$. It is like solving a puzzle; once you find the first piece, the rest falls into place.

Q: Why does the sign of the constant term change the result so drastically?

In quadratic equations, the constant term is the product of the roots. Changing the sign of the constant term changes the product of the roots, which in turn changes the value of the sum of squares $(a^2+b^2)$. Since our final expression relies heavily on these sums of squares, even a small change in the constant term propagates through the entire calculation.

Question

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 $x^2 - 20x + 2020$. 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

Sigma Percentile · Accepted Answer

### The Elegance of Symmetry: A JEE Masterclass

Welcome, future engineers. Today, we are going to dissect a problem that, at first glance, looks like a nightmare of algebra.

You see a quadratic equation, you see roots, and you see a long, intimidating expression. Your instinct might be to reach for the quadratic formula, but I want you to pause.

In the world of JEE Advanced, the most powerful tool in your arsenal is not calculation; it is observation.

### Phase 1: The Vieta Toolkit

Let us look at our two equations: $x^2 + 20x - 2020 = 0$ with roots $a, b$, and $x^2 - 20x + 2020 = 0$ with roots $c, d$.

Before we touch the target expression, we must extract the DNA of these equations using Vieta's formulas.

For the first equation, the sum of roots is $a + b = -20$ and the product is $ab = -2020$.

For the second, $c + d = 20$ and $cd = 2020$.

Notice the beautiful symmetry? The coefficients are just flipped. This is not a coincidence; it is a roadmap.

### Phase 2: The Expansion

Now, let us face the target expression: 
$$E = ac(a - c) + ad(a - d) + bc(b - c) + bd(b - d)$$

It looks like a mess, doesn't it? But let us expand it systematically.

We distribute the terms: 
$$E = a^2c - ac^2 + a^2d - ad^2 + b^2c - bc^2 + b^2d - bd^2$$

Now we have eight terms. It looks worse, but we have successfully broken the wall.

Now, we regroup. We collect terms with $a^2$ and $b^2$ together, and terms with $c^2$ and $d^2$ together.

This gives us: 
$$E = (a^2c + a^2d + b^2c + b^2d) - (ac^2 + ad^2 + bc^2 + bd^2)$$

### Phase 3: The Strategic Regrouping

This is where the magic happens. Look at the first group: $(a^2c + a^2d + b^2c + b^2d)$.

We can factor out $(c+d)$ from this group to get $(a^2 + b^2)(c + d)$.

Similarly, for the second group, we factor out $(a+b)$ to get $(c^2 + d^2)(a + b)$.

Our expression is now: 
$$E = (a^2 + b^2)(c + d) - (c^2 + d^2)(a + b)$$

We have reduced a complex polynomial into a simple product of sums and squares. This is the power of algebraic manipulation.

### Phase 4: The Final Calculation

We are almost there. We know $(a+b)$ and $(c+d)$, but we need $(a^2+b^2)$ and $(c^2+d^2)$.

We use the identity $x^2 + y^2 = (x+y)^2 - 2xy$.

For the first pair: 
$$a^2 + b^2 = (-20)^2 - 2(-2020) = 400 + 4040 = 4440$$

For the second pair: 
$$c^2 + d^2 = (20)^2 - 2(2020) = 400 - 4040 = -3640$$

Substituting these back into our expression: 
$$E = (4440)(20) - (-3640)(-20)$$

Calculating this, we get $88800 - 72800 = 16000$.

### Conclusion

Look at that! We started with a terrifying expression and ended with a clean, elegant integer.

The lesson here is simple: never rush to calculate. Always look for the structure, use Vieta's formulas to simplify, and trust the algebra.

You have the skills to solve any problem, provided you keep your cool and look for the symmetry. The final answer is **16000**.

Select Answer:

Visualized Solution

Quadratic Equations and Roots

Vieta's Formulas for Equation 1

Vieta's Formulas for Equation 2

The Target Expression

Expand the Expression

Group Terms Strategically

Factor Out Common Terms

Combine the Groups

Express Sum of Squares

Substitute Identities into E

Substitute Numerical Values

Simplify the Signs

Factor Out 20 and Compute

Final Result

The Sigma Insight: Relation Between Roots and Coefficients

The Elegance of Symmetry

Phase 1

Phase 2

Phase 3

Phase 4

Conclusion

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