Hello there, future engineers and mathematicians! Welcome to a truly foundational and fascinating topic in Quadratic Equations: the Relations between Roots and Coefficients. This concept is like the secret handshake between the solutions of an equation and the numbers that define it. Trust me, once you grasp this, you'll feel like you have superpowers when dealing with quadratic equations!

Let's dive in, assuming you're seeing this for the very first time.

### What's a Quadratic Equation Again?

Before we talk about roots and coefficients, let's quickly refresh our memory. A quadratic equation is a polynomial equation of degree 2. Its most general or standard form looks like this:

`ax² + bx + c = 0`

Here:
* `x` is the variable (the unknown we want to find).
* `a`, `b`, and `c` are constants, which we call coefficients.
* Crucially, `a` cannot be zero (because if `a = 0`, it would no longer be a quadratic equation, but a linear one!).
* The roots (or solutions) of a quadratic equation are the values of `x` that make the equation true. A quadratic equation always has two roots. These roots can be real or complex, and sometimes they can be equal.

For example, in `x² - 5x + 6 = 0`, `a=1`, `b=-5`, `c=6`. The roots are `x=2` and `x=3`. If you plug 2 or 3 into the equation, it will make the equation true (0 = 0).

### The Big Idea: A Secret Connection!

Imagine you have a magic box. You put some numbers (`a`, `b`, `c`) into it, and out come two other numbers (the roots, let's call them `α` (alpha) and `β` (beta)). What if there was a simple way to know the relationship between the numbers you put in and the numbers that came out, without even opening the box or doing complex calculations every time? That's exactly what these relations are all about!

These relations give us a direct link between the coefficients (`a`, `b`, `c`) and the roots (`α`, `β`). It's like a secret formula that tells us how the sum and product of the roots are directly dependent on the coefficients.

### Deriving the Fundamental Relations

Let's start with our standard quadratic equation:
`ax² + bx + c = 0`

We know from the Quadratic Formula that the two roots of this equation are given by:
`x = (-b ± √(b² - 4ac)) / (2a)`

Let's call our two roots `α` (alpha) and `β` (beta):
`α = (-b + √(b² - 4ac)) / (2a)`
`β = (-b - √(b² - 4ac)) / (2a)`

Now, let's see what happens when we sum these two roots and when we multiply them.

#### 1. Sum of the Roots (`α + β`)

Let's add `α` and `β`:

`α + β = [(-b + √(b² - 4ac)) / (2a)] + [(-b - √(b² - 4ac)) / (2a)]`

Since they have the same denominator, we can combine the numerators:

`α + β = (-b + √(b² - 4ac) - b - √(b² - 4ac)) / (2a)`

Notice how `+√(b² - 4ac)` and `-√(b² - 4ac)` cancel each other out!

`α + β = (-b - b) / (2a)`
`α + β = -2b / (2a)`

And finally, cancelling out the `2` from numerator and denominator:

`α + β = -b/a`

This is our first fundamental relation! The sum of the roots of a quadratic equation `ax² + bx + c = 0` is always equal to minus the coefficient of x divided by the coefficient of x².

#### 2. Product of the Roots (`α * β`)

Now, let's multiply `α` and `β`:

`α * β = [(-b + √(b² - 4ac)) / (2a)] * [(-b - √(b² - 4ac)) / (2a)]`

Remember the algebraic identity `(P + Q)(P - Q) = P² - Q²`? Here, `P = -b` and `Q = √(b² - 4ac)`.

So, the numerator becomes:
`(-b)² - (√(b² - 4ac))²`
`= b² - (b² - 4ac)`
`= b² - b² + 4ac`
`= 4ac`

And the denominator is `(2a) * (2a) = 4a²`.

Putting it all together:

`α * β = (4ac) / (4a²)`

Now, we can cancel out `4a` from the numerator and denominator:

`α * β = c/a`

This is our second fundamental relation! The product of the roots of a quadratic equation `ax² + bx + c = 0` is always equal to the constant term divided by the coefficient of x².

### Intuition Check: Why is this so useful?

Think of it this way:
If you have a quadratic equation `x² - (Sum of Roots)x + (Product of Roots) = 0`, then the coefficients directly tell you the sum and product of its roots.

For example, if the roots are `α` and `β`, then the quadratic equation can also be written in factored form as:
`(x - α)(x - β) = 0`

Let's expand this:
`x(x - β) - α(x - β) = 0`
`x² - βx - αx + αβ = 0`
`x² - (α + β)x + αβ = 0`

Compare this with the standard form `ax² + bx + c = 0`.
If `a=1`, then we can see:
`b = -(α + β)` => `α + β = -b`
`c = αβ` => `αβ = c`

What if `a` is not 1? Well, we can always divide the entire equation `ax² + bx + c = 0` by `a` (since `a ≠ 0`) to get:
`x² + (b/a)x + (c/a) = 0`

Now, comparing this to `x² - (α + β)x + αβ = 0`:
We can clearly see:
`-(α + β) = b/a` => `α + β = -b/a`
`αβ = c/a`

This alternative derivation gives you the same powerful result and reinforces the idea that these relations are *inherent* to the structure of quadratic equations.

Key Takeaway: The two fundamental relations are:
1. Sum of Roots: `α + β = -b/a`
2. Product of Roots: `α * β = c/a`

These are incredibly important, so make sure you memorize them!

### Let's Work Through Some Examples!

#### Example 1: Finding Sum and Product from an Equation

Problem: Find the sum and product of the roots for the quadratic equation `2x² + 5x - 12 = 0`.

Solution:
1. Identify coefficients:
In `2x² + 5x - 12 = 0`, we have:
`a = 2`
`b = 5`
`c = -12` (Don't forget the sign!)

2. Calculate the sum of roots:
`α + β = -b/a`
`α + β = -(5) / (2)`
`α + β = -5/2`

3. Calculate the product of roots:
`α * β = c/a`
`α * β = (-12) / (2)`
`α * β = -6`

So, for `2x² + 5x - 12 = 0`, the sum of roots is `-5/2` and the product of roots is `-6`.

#### Example 2: Forming an Equation from Given Roots

Problem: Form a quadratic equation whose roots are `3` and `-7`.

Solution:
1. Identify the roots:
Let `α = 3` and `β = -7`.

2. Calculate the sum of roots:
`Sum = α + β = 3 + (-7) = 3 - 7 = -4`

3. Calculate the product of roots:
`Product = α * β = 3 * (-7) = -21`

4. Use the general form:
The quadratic equation can be written as `x² - (Sum of Roots)x + (Product of Roots) = 0`.
`x² - (-4)x + (-21) = 0`
`x² + 4x - 21 = 0`

This is the required quadratic equation. You can quickly verify this by finding its roots using factorization or the quadratic formula; you'll find they are indeed 3 and -7!

#### Example 3: Using Relations to Find an Unknown Coefficient

Problem: If one root of the quadratic equation `x² - kx + 18 = 0` is `3`, find the value of `k` and the other root.

Solution:
1. Identify coefficients and known root:
In `x² - kx + 18 = 0`:
`a = 1`
`b = -k`
`c = 18`
Let `α = 3` be one root. Let the other root be `β`.

2. Use the product of roots relation:
We know `α * β = c/a`.
`3 * β = 18 / 1`
`3β = 18`
`β = 18 / 3`
`β = 6`
So, the other root is `6`.

3. Use the sum of roots relation to find k:
We know `α + β = -b/a`.
`3 + 6 = -(-k) / 1`
`9 = k`
So, `k = 9`.

The value of `k` is `9`, and the other root is `6`. This shows the power of these relations – we didn't even have to solve the quadratic formula completely to find the unknown!

### CBSE vs. JEE Focus

* For CBSE exams (and state boards), understanding these relations and being able to apply them in straightforward problems like the examples above is crucial. You might be asked to find sum/product, form an equation, or find an unknown coefficient/root.
* For JEE Mains & Advanced, this concept forms the absolute bedrock for much more complex problems. You'll use these relations constantly, but often in combination with other algebraic manipulations, properties of roots (like roots being reciprocals, or differing by a constant), and often in problems involving common roots of two quadratics or higher degree polynomial relations. The manipulations will be far more involved than just direct substitution. For example, you might be asked to find `α² + β²` or `1/α + 1/β` in terms of coefficients, which requires a bit more algebraic flair.

### Conclusion

The relations between roots and coefficients are more than just formulas; they are powerful tools that connect the visible parts of a quadratic equation (its coefficients) to its hidden solutions (its roots). Mastering `α + β = -b/a` and `α * β = c/a` will significantly simplify many problems and lay a strong foundation for your journey into higher-level algebra. Keep practicing, and these formulas will become second nature to you!

Welcome, aspiring mathematicians! Today, we are going to dive deep into a fundamental and incredibly powerful concept in quadratic equations: the Relations between Roots and Coefficients. This topic is not just a cornerstone for understanding quadratic equations but also forms the basis for solving complex problems efficiently, especially in competitive exams like JEE.

Think of a quadratic equation as a puzzle. The coefficients are the known pieces, and the roots are the hidden solutions. Our goal today is to understand the secret language that connects these pieces, allowing us to deduce properties of the solutions without even finding them explicitly.

---

### Understanding the Basics: What are Roots and Coefficients?

Before we jump into the relations, let's quickly recap. A general quadratic equation is written in the form:
$ax^2 + bx + c = 0$
where $a, b, c$ are real numbers, and $a
eq 0$.

* The numbers $a, b, c$ are called the coefficients of the quadratic equation. Specifically, $a$ is the coefficient of $x^2$, $b$ is the coefficient of $x$, and $c$ is the constant term.
* The values of $x$ that satisfy this equation are called its roots or solutions. A quadratic equation always has two roots, which can be real or complex, distinct or identical.

---

### Derivation of Relations between Roots and Coefficients

Let the two roots of the quadratic equation $ax^2 + bx + c = 0$ be $alpha$ and $eta$.
From the quadratic formula, we know that the roots are given by:
$x = frac{-b pm sqrt{b^2 - 4ac}}{2a}$

Let's define our two roots based on this formula:

1. $alpha = frac{-b + sqrt{b^2 - 4ac}}{2a}$


2. $eta = frac{-b - sqrt{b^2 - 4ac}}{2a}$

Now, let's derive the fundamental relations:

#### 1. Sum of the Roots ($alpha + eta$)

Let's add $alpha$ and $eta$:
$alpha + eta = left( frac{-b + sqrt{b^2 - 4ac}}{2a}
ight) + left( frac{-b - sqrt{b^2 - 4ac}}{2a}
ight)$
$alpha + eta = frac{-b + sqrt{b^2 - 4ac} - b - sqrt{b^2 - 4ac}}{2a}$
$alpha + eta = frac{-2b}{2a}$
$oxed{alpha + eta = -frac{b}{a}}$

This is a crucial relation: The sum of the roots of a quadratic equation is equal to the negative of the coefficient of $x$ divided by the coefficient of $x^2$.

#### 2. Product of the Roots ($alpha eta$)

Now, let's multiply $alpha$ and $eta$:
$alpha eta = left( frac{-b + sqrt{b^2 - 4ac}}{2a}
ight) left( frac{-b - sqrt{b^2 - 4ac}}{2a}
ight)$
This looks like $(P+Q)(P-Q)$ which simplifies to $P^2 - Q^2$, where $P = -b$ and $Q = sqrt{b^2 - 4ac}$.
$alpha eta = frac{(-b)^2 - (sqrt{b^2 - 4ac})^2}{(2a)^2}$
$alpha eta = frac{b^2 - (b^2 - 4ac)}{4a^2}$
$alpha eta = frac{b^2 - b^2 + 4ac}{4a^2}$
$alpha eta = frac{4ac}{4a^2}$
$oxed{alpha eta = frac{c}{a}}$

This is another crucial relation: The product of the roots of a quadratic equation is equal to the constant term divided by the coefficient of $x^2$.

#### 3. Difference of the Roots ($|alpha - eta|$)

The difference between the roots is also important. Let's find $alpha - eta$:
$alpha - eta = left( frac{-b + sqrt{b^2 - 4ac}}{2a}
ight) - left( frac{-b - sqrt{b^2 - 4ac}}{2a}
ight)$
$alpha - eta = frac{-b + sqrt{b^2 - 4ac} + b + sqrt{b^2 - 4ac}}{2a}$
$alpha - eta = frac{2sqrt{b^2 - 4ac}}{2a}$
$alpha - eta = frac{sqrt{b^2 - 4ac}}{a}$

Since the order of roots can be arbitrary, we usually refer to the absolute difference:
$oxed{|alpha - eta| = frac{sqrt{b^2 - 4ac}}{|a|} = frac{sqrt{D}}{|a|}}$, where $D = b^2 - 4ac$ is the discriminant.

---

### Formation of a Quadratic Equation from its Roots

If we know the roots $alpha$ and $eta$ of a quadratic equation, we can reconstruct the equation.
If $alpha$ and $eta$ are the roots, then $(x - alpha)$ and $(x - eta)$ must be factors of the quadratic polynomial.
Therefore, the quadratic equation can be written as:
$(x - alpha)(x - eta) = 0$
Expanding this, we get:
$x^2 - eta x - alpha x + alpha eta = 0$
$x^2 - (alpha + eta)x + alpha eta = 0$

Substituting the sum and product of roots:
$oxed{x^2 - ( ext{Sum of roots})x + ( ext{Product of roots}) = 0}$

This is an incredibly useful form for quickly constructing a quadratic equation when its roots are known or when we know the sum and product of its roots. Note that this gives the monic form (coefficient of $x^2$ is 1). If the original equation was $ax^2 + bx + c = 0$, then it would be $a(x^2 - (alpha+eta)x + alphaeta) = 0$.

---

### CBSE vs. JEE Focus: Symmetric Functions of Roots

While CBSE typically focuses on the basic sum, product, and formation of equations, JEE often tests your ability to manipulate expressions involving roots using these fundamental relations. This involves symmetric functions of roots.

A function $f(alpha, eta)$ is called a symmetric function if its value remains unchanged when $alpha$ and $eta$ are interchanged, i.e., $f(alpha, eta) = f(eta, alpha)$.
All symmetric functions of $alpha$ and $eta$ can be expressed in terms of $(alpha + eta)$ and $(alpha eta)$.

Here are some common symmetric functions and their expressions:

1. 
   $alpha^2 + eta^2$
   
   We know $(alpha + eta)^2 = alpha^2 + 2alphaeta + eta^2$.
   
   So, $oxed{alpha^2 + eta^2 = (alpha + eta)^2 - 2alphaeta}$
   


2. 
   $alpha^3 + eta^3$
   
   We know $(alpha + eta)^3 = alpha^3 + 3alpha^2eta + 3alphaeta^2 + eta^3 = alpha^3 + eta^3 + 3alphaeta(alpha + eta)$.
   
   So, $oxed{alpha^3 + eta^3 = (alpha + eta)^3 - 3alphaeta(alpha + eta)}$
   


3. 
   $alpha - eta$ (or $eta - alpha$)
   
   We know $(alpha - eta)^2 = (alpha + eta)^2 - 4alphaeta$.
   
   So, $oxed{|alpha - eta| = sqrt{(alpha + eta)^2 - 4alphaeta}}$ (This is consistent with $frac{sqrt{D}}{|a|}$)
   


4. 
   $frac{1}{alpha} + frac{1}{eta}$
   
   $oxed{frac{1}{alpha} + frac{1}{eta} = frac{alpha + eta}{alphaeta}}$
   


5. 
   $frac{alpha}{eta} + frac{eta}{alpha}$
   
   $frac{alpha}{eta} + frac{eta}{alpha} = frac{alpha^2 + eta^2}{alphaeta}$
   
   Substitute $alpha^2 + eta^2 = (alpha + eta)^2 - 2alphaeta$:
   
   $oxed{frac{alpha}{eta} + frac{eta}{alpha} = frac{(alpha + eta)^2 - 2alphaeta}{alphaeta}}$
   

Mastering these transformations is key for JEE problems where direct calculation of roots might be tedious or impossible.

---

### Important Observations and Special Cases

Understanding these relations allows us to make quick deductions about the nature of roots based on coefficients.

1. 
   If $b=0$:
   The equation becomes $ax^2 + c = 0 implies x^2 = -c/a$.
   The roots are $alpha = sqrt{-c/a}$ and $eta = -sqrt{-c/a}$.
   In this case, $alpha + eta = 0$. Since $alpha + eta = -b/a$, it implies $-b/a = 0$, so $b=0$.
   This means the roots are equal in magnitude but opposite in sign. (e.g., $x^2 - 4 = 0 implies x = pm 2$)
   


2. 
   If $c=0$:
   The equation becomes $ax^2 + bx = 0 implies x(ax + b) = 0$.
   The roots are $x=0$ and $x=-b/a$.
   In this case, $alpha eta = 0$. Since $alpha eta = c/a$, it implies $c/a = 0$, so $c=0$.
   This means one of the roots is zero. (e.g., $x^2 + 3x = 0 implies x(x+3)=0 implies x=0, -3$)
   


3. 
   If $c=a$:
   Then $alpha eta = c/a = 1$.
   This means the roots are reciprocals of each other, i.e., $eta = 1/alpha$. (e.g., $2x^2 - 5x + 2 = 0 implies (2x-1)(x-2)=0 implies x=1/2, 2$)
   


4. 
   If $a+b+c=0$:
   Substitute $x=1$ into the equation: $a(1)^2 + b(1) + c = a+b+c$.
   If $a+b+c=0$, then $x=1$ is a root.
   The other root, $eta$, can be found using the product of roots: $1 cdot eta = c/a implies eta = c/a$.
   So, if the sum of coefficients is zero, one root is 1 and the other is $c/a$.
   


5. 
   If $a-b+c=0$:
   Substitute $x=-1$ into the equation: $a(-1)^2 + b(-1) + c = a-b+c$.
   If $a-b+c=0$, then $x=-1$ is a root.
   The other root, $eta$, can be found using the product of roots: $(-1) cdot eta = c/a implies eta = -c/a$.
   So, if $a-b+c=0$, one root is -1 and the other is $-c/a$.
   

---

### Examples for Application

Let's put these relations to use with some examples.

#### Example 1: Basic Application

Given the quadratic equation $3x^2 - 5x + 2 = 0$. Find the sum and product of its roots without solving for the roots.

Solution:
Here, $a=3$, $b=-5$, $c=2$.
Let the roots be $alpha$ and $eta$.
Sum of roots: $alpha + eta = -frac{b}{a} = -frac{(-5)}{3} = frac{5}{3}$
Product of roots: $alpha eta = frac{c}{a} = frac{2}{3}$

#### Example 2: Forming a Quadratic Equation

Form a quadratic equation whose roots are $3 + sqrt{2}$ and $3 - sqrt{2}$.

Solution:
Let $alpha = 3 + sqrt{2}$ and $eta = 3 - sqrt{2}$.
Sum of roots: $alpha + eta = (3 + sqrt{2}) + (3 - sqrt{2}) = 6$
Product of roots: $alpha eta = (3 + sqrt{2})(3 - sqrt{2}) = 3^2 - (sqrt{2})^2 = 9 - 2 = 7$

Using the formula $x^2 - ( ext{Sum of roots})x + ( ext{Product of roots}) = 0$:
The equation is $x^2 - 6x + 7 = 0$.

#### Example 3: Symmetric Functions (JEE Level)

If $alpha$ and $eta$ are the roots of the equation $2x^2 - 7x + 4 = 0$, find the value of $alpha^2 + eta^2$.

Solution:
First, identify the coefficients: $a=2, b=-7, c=4$.
Now, find the sum and product of the roots:
$alpha + eta = -frac{b}{a} = -frac{(-7)}{2} = frac{7}{2}$
$alpha eta = frac{c}{a} = frac{4}{2} = 2$

We need to find $alpha^2 + eta^2$. Using our symmetric function formula:
$alpha^2 + eta^2 = (alpha + eta)^2 - 2alphaeta$
Substitute the values:
$alpha^2 + eta^2 = left(frac{7}{2}
ight)^2 - 2(2)$
$alpha^2 + eta^2 = frac{49}{4} - 4$
$alpha^2 + eta^2 = frac{49}{4} - frac{16}{4}$
$alpha^2 + eta^2 = frac{33}{4}$

#### Example 4: Advanced Problem (JEE Type)

If the roots of the equation $x^2 - px + q = 0$ are $alpha$ and $eta$, and the roots of the equation $x^2 - ax + b = 0$ are $alpha^2$ and $eta^2$, then find the relation between $a, b, p, q$.

Solution:
For $x^2 - px + q = 0$:
Sum of roots: $alpha + eta = p$
Product of roots: $alpha eta = q$

For $x^2 - ax + b = 0$:
Sum of roots: $alpha^2 + eta^2 = a$
Product of roots: $alpha^2 eta^2 = b$

We know that $alpha^2 + eta^2 = (alpha + eta)^2 - 2alphaeta$.
Substitute the values from the first equation into this identity:
$a = (p)^2 - 2(q)$
So, $a = p^2 - 2q$.

Also, we know that $alpha^2 eta^2 = (alpha eta)^2$.
Substitute the values:
$b = (q)^2$
So, $b = q^2$.

The relations are $a = p^2 - 2q$ and $b = q^2$. These relate the coefficients of the two quadratic equations.

---

### Extension to Higher Degree Polynomials (Briefly for JEE Advanced)

The concept of relations between roots and coefficients extends to polynomials of higher degrees. For a general polynomial equation of degree $n$:
$a_n x^n + a_{n-1} x^{n-1} + dots + a_1 x + a_0 = 0$
with roots $r_1, r_2, dots, r_n$:

Relation	Formula
Sum of roots taken one at a time ($sum r_i$)	$r_1 + r_2 + dots + r_n = -frac{a_{n-1}}{a_n}$
Sum of products of roots taken two at a time ($sum r_i r_j$)	$r_1 r_2 + r_1 r_3 + dots + r_{n-1} r_n = frac{a_{n-2}}{a_n}$
Sum of products of roots taken three at a time ($sum r_i r_j r_k$)	$r_1 r_2 r_3 + dots = -frac{a_{n-3}}{a_n}$
...	...
Product of all roots ($r_1 r_2 dots r_n$)	$(-1)^n frac{a_0}{a_n}$

For a cubic equation $ax^3 + bx^2 + cx + d = 0$ with roots $alpha, eta, gamma$:

• $alpha + eta + gamma = -frac{b}{a}$


• $alphaeta + etagamma + gammaalpha = frac{c}{a}$


• $alphaetagamma = -frac{d}{a}$

This pattern (alternating signs for coefficients divided by the leading coefficient) is known as Vieta's Formulas and is extremely important for higher-level polynomial problems in JEE.

---

### Conclusion

The relations between roots and coefficients are more than just formulas; they are powerful tools that allow us to understand the intrinsic properties of a quadratic equation's solutions without direct computation. For CBSE students, mastering the sum, product, and forming equations is key. For JEE aspirants, extending this understanding to symmetric functions, special cases, and even higher-degree polynomials is absolutely essential for tackling complex problems efficiently. Keep practicing, and these relations will become second nature!

Understanding the relations between roots and coefficients is fundamental for solving quadratic equations efficiently and is a high-yield topic for JEE Main. Here are some mnemonics and shortcuts to master these concepts.

Mnemonics for Quadratic Equations ($ax^2 + bx + c = 0$)

Let $alpha$ and $eta$ be the roots of the quadratic equation $ax^2 + bx + c = 0$.

• Sum of Roots: $alpha + eta = -b/a$
  
  
  
  • Mnemonic: "Sum Is Negative By A" (S.I.N.B.A. $
    ightarrow$ Sum, Is, Negative, B, A). This reminds you of the negative sign before 'b'.
  
  
  • Alternate: Think of the 'b' term as 'opposite'. The sum of roots is the opposite of the 'b' coefficient divided by 'a'.
  
  
  
  


• Product of Roots: $alpha eta = c/a$
  
  
  
  • Mnemonic: "Product Is Positive Constant A" (P.I.P.C.A. $
    ightarrow$ Product, Is, Positive, C, A). This emphasizes that the constant 'c' is involved and the sign is positive.
  
  
  • Alternate: The product is "Clean" (no negative sign) and involves the Constant term 'c'.
  
  
  
  


• Difference of Roots: $|alpha - eta| = sqrt{D}/|a|$ where $D = b^2 - 4ac$
  
  
  
  • Mnemonic: "Difference Depends on Discriminant." This helps remember that the discriminant 'D' is crucial here.
  
  
  • Alternate: "Root of Delta over Absolute A." (R-D-A-A). Remember the absolute value for 'a' to ensure the difference is non-negative.
  
  
  
  

Shortcuts & Practical Tips (JEE Focused)

• Standard Form First: Always ensure the quadratic equation is in the standard form $ax^2 + bx + c = 0$. If it's $ax^2 + bx = -c$, rearrange it before applying the formulas. Common Mistake: Incorrectly identifying 'c' if not in standard form.


• Check 'a': If the coefficient of $x^2$ (i.e., 'a') is not 1, remember to divide by 'a' in both sum and product formulas. Many sign errors occur when 'a' is negative.


• Symmetric Roots:
  
  
  
  • If roots are reciprocal ($alpha = 1/eta$), then $alpha eta = 1 Rightarrow c/a = 1 Rightarrow mathbf{c=a}$.
  
  
  • If roots are opposite in sign ($alpha = -eta$), then $alpha + eta = 0 Rightarrow -b/a = 0 Rightarrow mathbf{b=0}$. (This implies the roots are $k$ and $-k$).
  
  
  
  


• Forming a New Quadratic Equation: If $alpha$ and $eta$ are the roots of an equation, the equation can be written as $x^2 - (alpha+eta)x + alphaeta = 0$.
  
  
  
  • Shortcut: If the roots of a new equation are $f(alpha)$ and $f(eta)$, sometimes a simple transformation can be applied. For example, if roots are $alpha+k, eta+k$, replace $x$ with $(x-k)$ in the original equation.
  
  
  
  


• Higher Degree Equations (JEE Advanced Relevance):
  

  For a polynomial $a_n x^n + a_{n-1} x^{n-1} + dots + a_1 x + a_0 = 0$ with roots $r_1, r_2, dots, r_n$:

  
  
  
  
  • Sum of roots (taken one at a time): $sum r_i = -a_{n-1}/a_n$
  
  
  • Sum of roots taken two at a time: $sum r_i r_j = a_{n-2}/a_n$
  
  
  • Sum of roots taken three at a time: $sum r_i r_j r_k = -a_{n-3}/a_n$
  
  
  • ...and so on.
  
  
  • Product of all roots: $r_1 r_2 dots r_n = (-1)^n cdot a_0/a_n$
  
  
  • Mnemonic for Signs: The sign of the $k^{th}$ sum (sum of products of $k$ roots) is $(-1)^k$. Always divide by the leading coefficient $a_n$.
  
  
  
  

JEE Tip: Practicing problems with these relations is key. Focus on minimizing sign errors and quickly identifying 'a', 'b', and 'c' from any given equation.

Understanding the relations between roots and coefficients is fundamental for solving quadratic equation problems efficiently in both CBSE board exams and JEE Main.

Core Relations (Quick Recap)

For a quadratic equation $ax^2 + bx + c = 0$ (where $a
eq 0$), let the roots be $alpha$ and $eta$.

• Sum of roots: $alpha + eta = -frac{b}{a}$


• Product of roots: $alpha eta = frac{c}{a}$


• Forming a quadratic equation: If roots are $alpha$ and $eta$, the equation is $x^2 - (alpha + eta)x + alpha eta = 0$. Tip: Always ensure the coefficient of $x^2$ is $1$ before applying this format.

Key Observations & Shortcuts

• One root is the negative of the other: If $alpha = -eta$, then $alpha + eta = 0$. This implies $-frac{b}{a} = 0$, so $b=0$. (The coefficient of $x$ is zero).


• One root is the reciprocal of the other: If $alpha = frac{1}{eta}$, then $alpha eta = 1$. This implies $frac{c}{a} = 1$, so $c=a$. (The coefficient of $x^2$ is equal to the constant term).


• One root is zero: If $alpha = 0$, then $alpha eta = 0$. This implies $frac{c}{a} = 0$, so $c=0$. (The constant term is zero).


• Roots are equal: If $alpha = eta$, then the discriminant $D = b^2 - 4ac = 0$.


• Both roots are zero: This occurs if $b=0$ and $c=0$.


• Roots are rational: This happens if $D$ is a perfect square (and $a, b, c$ are rational).


• Roots are irrational: If $D$ is not a perfect square (and $a, b, c$ are rational). Irrational roots always occur in conjugate pairs ($p pm sqrt{q}$).


• Roots are complex: If $D < 0$. Complex roots always occur in conjugate pairs ($p pm iq$).

Symmetric Functions of Roots

Many JEE problems require expressing higher powers or combinations of roots in terms of $alpha + eta$ and $alpha eta$. Remember these standard identities:

• $alpha^2 + eta^2 = (alpha + eta)^2 - 2alpha eta$


• $alpha^3 + eta^3 = (alpha + eta)^3 - 3alpha eta (alpha + eta)$


• $alpha^4 + eta^4 = (alpha^2 + eta^2)^2 - 2(alpha eta)^2$


• $frac{1}{alpha} + frac{1}{eta} = frac{alpha + eta}{alpha eta}$


• $frac{alpha}{eta} + frac{eta}{alpha} = frac{alpha^2 + eta^2}{alpha eta} = frac{(alpha+eta)^2 - 2alphaeta}{alphaeta}$


• $(alpha - eta)^2 = (alpha + eta)^2 - 4alpha eta$

JEE Tip: For higher powers, you can also use the fact that roots satisfy the equation. If $alpha$ is a root of $ax^2+bx+c=0$, then $aalpha^2+balpha+c=0$, which implies $aalpha^2 = -balpha - c$. This can be used to reduce the power of roots recursively.

Generalization to Higher Degree Polynomials (Vieta's Formulas)

While quadratics are the primary focus, be aware of the generalization. For a polynomial $a_nx^n + a_{n-1}x^{n-1} + dots + a_1x + a_0 = 0$ with roots $r_1, r_2, dots, r_n$:

• Sum of roots: $sum r_i = -frac{a_{n-1}}{a_n}$


• Sum of products of roots taken two at a time: $sum_{i
  
• Sum of products of roots taken three at a time: $sum_{i
  
• ...and so on, alternating signs.


• Product of roots: $r_1 r_2 dots r_n = (-1)^n frac{a_0}{a_n}$

Practical Advice: Always simplify the quadratic equation to its standard form ($ax^2+bx+c=0$) before applying the relations. Pay close attention to the signs of coefficients, especially for 'b'.

While the study of relations between roots and coefficients might seem abstract, particularly Vieta's formulas, their underlying principles find practical utility in various real-world scenarios, especially in fields that involve modeling and optimization using quadratic functions. Understanding these relationships allows engineers, scientists, and developers to analyze system behavior without necessarily needing to calculate the exact roots explicitly.

Key Real-World Applications

• 
  Engineering Design and Optimization:
  
  
  
  • In electrical engineering, designing filters or resonant circuits often involves quadratic equations. The nature and values of roots (e.g., frequencies or damping factors) dictate the circuit's performance. Vieta's formulas can help relate component values (coefficients) to desired frequency responses or stability criteria (root properties).
  
  
  • Mechanical engineering problems, such as optimizing beam deflection or designing vibrating systems, can lead to quadratic equations. The sum or product of roots might represent critical parameters for stability or efficient operation.
  
  
  
  


• 
  Physics and Trajectory Analysis:
  
  
  
  • For projectile motion, the height of an object often follows a quadratic path. If `h(t) = at^2 + bt + c`, the roots represent the times when the object hits a certain height (e.g., the ground). The sum and product of these roots can provide information about total flight time or horizontal range based on initial velocity and height, without needing to solve the quadratic equation directly. This is useful for quickly assessing scenarios.
  
  
  
  


• 
  Computer Graphics and Game Development:
  
  
  
  • In ray tracing, a fundamental technique for rendering realistic 3D graphics, one often needs to determine if a ray (a line) intersects with a geometric primitive like a sphere or an ellipsoid. This intersection problem invariably reduces to solving a quadratic equation.
  
  
  • The nature of the roots (real and distinct, real and equal, or complex) directly tells the graphics engine whether the ray hits the object, grazes it, or misses it entirely. The sum and product of the roots can help efficiently calculate the two intersection points or their midpoint, which is crucial for determining surface normals and lighting effects.
  
  
  
  


• 
  Economics and Business Modeling:
  
  
  
  • In economic models, cost functions, revenue functions, or profit functions are sometimes modeled as quadratic equations. For instance, determining break-even points or points of maximum profit/minimum cost often involves finding roots of quadratic equations. Vieta's formulas can help analyze the relationship between input parameters (coefficients) and these critical points.
  
  
  
  

Illustrative Example: Ray-Sphere Intersection (Computer Graphics)

Consider a ray originating from point O with direction vector D, represented as P(t) = O + tD. We want to find its intersection with a sphere centered at C with radius R, described by |X - C|^2 = R^2.

Substituting X = P(t) into the sphere equation, we get a quadratic equation in t:

A t^2 + B t + E = 0

where A = |D|^2, B = 2 D ⋅ (O - C), and E = |O - C|^2 - R^2.

• If the discriminant Δ = B^2 - 4AE < 0 (complex roots), the ray misses the sphere.


• If Δ = 0 (repeated root), the ray grazes the sphere at one point.


• If Δ > 0 (two distinct real roots t1, t2), the ray intersects the sphere at two points.

Using Vieta's formulas, we have:

• Sum of roots: t1 + t2 = -B/A


• Product of roots: t1 * t2 = E/A

These relations are extremely useful:

• Without calculating t1 and t2 explicitly, their sum and product provide information about the average intersection distance or the relative position of intersection points along the ray.


• For surface normal calculation, the actual intersection points are needed, but the properties of roots help in early culling (missing the sphere) or in optimizing subsequent calculations.

This example highlights how understanding the relationship between the coefficients (which contain information about the ray and sphere) and the roots (which represent intersection times) is fundamental to efficient algorithm design in practical applications like computer graphics.

Understanding the relations between roots and coefficients is fundamental to solving problems involving quadratic equations. While the concept itself is straightforward, analogies can help solidify your intuition about *why* these relations exist and *how* they link the different parts of an equation. Think of it as decoding the DNA of a polynomial.

Analogy: The Recipe and its Key Ingredients

Imagine a quadratic equation $ax^2 + bx + c = 0$ as a delicious cake, and its roots, $alpha$ and $eta$, as the essential, distinct ingredients that define its unique flavour and structure (e.g., flour and sugar).

• 
  The "Cake" (Quadratic Equation): Your fully baked quadratic equation, $ax^2 + bx + c = 0$, is the final product.
  


• 
  The "Key Ingredients" (Roots): The roots $alpha$ and $eta$ are the fundamental building blocks. Just as flour and sugar are crucial for a cake, $alpha$ and $eta$ are the specific values that make the equation true.
  


• 
  The "Recipe Information" (Coefficients): The coefficients $a, b, c$ are not the ingredients themselves, but rather specific "summary instructions" or "derived properties" from how these ingredients interact in the recipe. They tell you about the ingredients without explicitly naming them individually.
  
  
  
  • Sum of Roots ($alpha + eta = -b/a$): This is like knowing the "total combined weight" of the flour and sugar. You might not know the individual weight of flour or sugar, but you know their sum. This specific sum directly influences the cake's overall sweetness or texture.
  
  
  • Product of Roots ($alphaeta = c/a$): This could be analogous to the "ratio of flour to sugar," or some other combined metric. It's another crucial piece of information derived from the ingredients that dictates a specific aspect of the cake's final state (e.g., how moist or dense it will be, or how quickly it browns).
  
  
  
  

The beauty of this analogy is that it highlights the powerful interdependence:

• If you know the ingredients ($alpha, eta$), you can easily reconstruct the "recipe information" (coefficients $b, c$) for the cake.


• Conversely, if you know the "recipe information" (coefficients $b, c$), you can infer important properties about the "ingredients" ($alpha, eta$) even if you don't know them individually.

JEE Relevance:

This analogy is particularly useful for JEE because it reinforces that the coefficients are not arbitrary numbers but are intricately linked to the roots. Many problems test your ability to move between the "ingredients" and the "recipe information":

• Forming Equations: If you're given roots, you can immediately write down the equation using these relations.


• Finding Unknowns: If you're given some conditions about the roots (e.g., one root is double the other, or roots are reciprocals), you can use the sum and product relations to set up equations and find unknown coefficients or specific root values.


• Generalization: This concept extends seamlessly to higher-degree polynomials (cubic, quartic), where more "summary instructions" (coefficients) are needed to describe the interplay of more "ingredients" (roots).

Mathematical Concept	Analogy: The Cake Recipe
Quadratic Equation ($ax^2 + bx + c = 0$)	The final "Cake"
Roots ($alpha, eta$)	The "Key Ingredients" (e.g., flour, sugar)
Coefficients ($a, b, c$)	"Recipe Information" / "Derived Properties"
Sum of Roots ($alpha + eta = -b/a$)	"Total combined weight of flour and sugar"
Product of Roots ($alphaeta = c/a$)	"Ratio or combined effect of ingredients"

Keep these connections in mind as you tackle problems; they simplify complex scenarios into manageable pieces.

Prerequisites for Relations between Roots and Coefficients

To effectively understand the relations between the roots and coefficients of a quadratic equation, a strong grasp of the following fundamental concepts is essential. These form the bedrock upon which the more advanced topic is built.

• 
  Definition and Standard Form of a Quadratic Equation:
  
  
  
  • You must be familiar with what constitutes a quadratic equation, i.e., an equation of the form ax² + bx + c = 0, where 'a', 'b', and 'c' are real numbers and a ≠ 0. Understanding that 'a', 'b', and 'c' are the coefficients is crucial.
  
  
  
  


• 
  Roots (or Solutions) of a Quadratic Equation:
  
  
  
  • A clear understanding that the 'roots' are the values of the variable (usually 'x') that satisfy the equation. A quadratic equation always has exactly two roots (which may be real or complex, distinct or identical).
  
  
  
  


• 
  Quadratic Formula (Sridharacharya Formula):
  
  
  
  • This is arguably the most critical prerequisite. You should be proficient in using the quadratic formula: x = [-b ± √(b² - 4ac)] / 2a. This formula explicitly links the roots directly to the coefficients 'a', 'b', and 'c', making it a direct precursor to understanding their general relations.
  
  
  • JEE Focus: Mastery of this formula is not just about finding roots but understanding its structural implication for root-coefficient relationships.
  
  
  
  


• 
  Nature of Roots (Discriminant):
  
  
  
  • Knowledge of the discriminant, D = b² - 4ac, and how its value determines the nature of the roots (real and distinct, real and equal, or complex conjugates) is important. This concept is derived directly from the quadratic formula and is intrinsically linked to the coefficients.
  
  
  
  


• 
  Basic Algebraic Manipulation:
  
  
  
  • Proficiency in fundamental algebraic operations such as addition, subtraction, multiplication, and division of expressions, as well as factorization and expansion of binomials, is necessary for deriving and working with the relations.
  
  
  
  

Before diving into relations like sum and product of roots, ensure you can comfortably define, identify, and find the roots of any given quadratic equation using the quadratic formula. This foundational knowledge will make understanding the relationships between roots and coefficients straightforward and intuitive.

Understanding the "Relations between roots and coefficients" for a quadratic equation (ax² + bx + c = 0) is a foundational concept in Algebra. This section outlines other topics that are directly related, build upon, or are prerequisites for a complete grasp of this concept.

Here are the key related topics:

• Quadratic Equations: Definition & Basic Solutions
  
  
  
  • Relevance: This is a fundamental prerequisite. Before understanding relations, one must know what a quadratic equation is and how to find its roots (e.g., factorization, quadratic formula).
  
  
  • JEE/CBSE: Essential for both.
  
  
  
  


• Nature of Roots
  
  
  
  • Relevance: The discriminant (Δ = b² - 4ac), which determines the nature of roots (real & distinct, real & equal, imaginary), is derived directly from the coefficients. This topic links the coefficients to the qualitative properties of the roots.
  
  
  • JEE/CBSE: Highly important for both, often tested in conjunction with root-coefficient relations.
  
  
  
  


• Formation of a Quadratic Equation with Given Roots
  
  
  
  • Relevance: This is a direct application. If the roots (α and β) are given, the equation can be formed as x² - (α+β)x + αβ = 0, which directly uses the sum and product of roots.
  
  
  • JEE/CBSE: Crucial for both, forms the basis for many problem types.
  
  
  
  


• Symmetric Functions of Roots
  
  
  
  • Relevance: Expressions involving roots (e.g., α² + β², α³ + β³, 1/α + 1/β) that remain unchanged if roots are swapped are called symmetric functions. These can always be expressed in terms of the elementary symmetric polynomials (α+β and αβ), which are directly related to the coefficients.
  
  
  • JEE Emphasis: This is a very common topic in JEE Main and Advanced, often requiring manipulation of root-coefficient relations.
  
  
  
  


• Common Roots
  
  
  
  • Relevance: When two quadratic equations share one or both roots, conditions involving their coefficients can be established. These conditions frequently utilize the relations between roots and coefficients of each equation.
  
  
  • JEE Emphasis: A significant topic for JEE, often involving system of equations derived from root relations.
  
  
  
  


• Location of Roots (or Interval of Roots)
  
  
  
  • Relevance: This topic deals with determining the range or interval in which the roots of a quadratic equation lie without actually finding the roots. It involves analyzing the discriminant, the value of the function at certain points, and the axis of symmetry, all of which are dependent on the coefficients.
  
  
  • JEE Emphasis: Highly important for JEE Advanced, sometimes for JEE Main.
  
  
  
  


• Transformation of Equations
  
  
  
  • Relevance: If the roots of a given equation are α and β, this topic involves forming a new quadratic equation whose roots are some function of α and β (e.g., α+k, α², 1/α). This process heavily relies on finding the sum and product of the new roots using the original root-coefficient relations.
  
  
  • JEE Emphasis: Common in JEE problems.
  
  
  
  


• Polynomial Equations of Higher Degree (Cubic, Biquadratic, etc.)
  
  
  
  • Relevance: The concept of relations between roots and coefficients generalizes to polynomials of any degree. For example, for a cubic equation ax³ + bx² + cx + d = 0, similar relations exist for the sum of roots, sum of products of roots taken two at a time, and product of all roots.
  
  
  • JEE Emphasis: The generalization to cubic and biquadratic equations is important for JEE.
  
  
  
  

Mastering the relations between roots and coefficients provides a powerful tool for solving a wide array of problems in quadratic equations and polynomials, making it a cornerstone for higher algebraic concepts.

Common Exam Traps: Relations Between Roots and Coefficients

Understanding the relations between roots and coefficients is fundamental, but exams often set traps to test your attention to detail and conceptual clarity. Be vigilant for the following common pitfalls:

• Trap 1: Sign Errors in Sum of Roots
  

  A very common mistake is to misremember the sign in the sum of roots. For a quadratic equation $ax^2 + bx + c = 0$, the sum of roots is $alpha + eta = mathbf{-b/a}$, NOT $b/a$. This seemingly small error can propagate through an entire problem, leading to incorrect answers. Always double-check the sign, especially when $b$ itself is negative or involves a parameter.

  
  



• Trap 2: Forgetting the Discriminant Condition for Real Roots
  

  Many problems related to root relations implicitly or explicitly state that the roots are "real." If this condition is mentioned, or implied (e.g., finding the range of a parameter for real roots), you must apply the discriminant condition $D = b^2 - 4ac ge 0$. Failing to do so can lead to an incorrect range of values for the parameter or overlooking a crucial constraint. For JEE Advanced, this is frequently tested with complex conditions.

  
  



• Trap 3: Misinterpreting Conditions on Roots
  

  Students often confuse or misinterpret specific conditions given for the roots:

  
  
  
  
  • Roots are opposite in sign: This implies $alphaeta < 0$, which means $c/a < 0$. No specific condition on the sum of roots.
  
  
  • Roots are equal in magnitude but opposite in sign: This implies $alpha + eta = 0$ (so $-b/a = 0 implies b=0$) AND $alphaeta < 0$ (so $c/a < 0$). Many students only remember $b=0$ and forget the product condition, which ensures distinct real roots. This is a classic trap in JEE Main.
  
  
  • One root is the reciprocal of the other: This implies $alpha = 1/eta$, so $alphaeta = 1$, which means $c/a = 1 implies c=a$.
  
  
  
  



• Trap 4: Variable Coefficient of $x^2$
  

  If the coefficient of $x^2$ (i.e., 'a') involves a parameter (e.g., $(k-1)x^2 + dots = 0$), always consider the case when this coefficient becomes zero. If $a=0$, the equation is no longer quadratic but linear, and it will have only one root, not two. Questions often ask for a parameter range for a "quadratic equation" or "an equation having two roots," which implicitly means $a
  e 0$. This distinction is vital for JEE problem-solving.

  
  



• Trap 5: Algebraic Manipulation Errors for Symmetric Expressions
  

  When asked to find expressions like $alpha^2 + eta^2$, $alpha^3 + eta^3$, or $frac{1}{alpha} + frac{1}{eta}$, students sometimes make algebraic errors. Remember the fundamental identities:

  
  
  
  
  • $alpha^2 + eta^2 = (alpha+eta)^2 - 2alphaeta$
  
  
  • $alpha^3 + eta^3 = (alpha+eta)(alpha^2 - alphaeta + eta^2) = (alpha+eta)[(alpha+eta)^2 - 3alphaeta]$
  
  
  • $frac{1}{alpha} + frac{1}{eta} = frac{alpha+eta}{alphaeta}$
  
  
  
  

  Practice these manipulations to avoid simple but costly mistakes in calculations.

  
  

Stay focused and practice rigorously to identify and avoid these common traps. Success in exams often comes from avoiding silly mistakes as much as from knowing the concepts.

Key Takeaways: Relations Between Roots and Coefficients

Understanding the relationship between the roots and coefficients of a quadratic equation is fundamental for solving a wide range of problems in algebra. These relationships allow you to determine properties of the roots without actually solving the equation, which is particularly useful in competitive exams like JEE Main.

Standard Quadratic Equation:

A general quadratic equation is given by $mathbf{ax^2 + bx + c = 0}$, where $a
eq 0$. Let $alpha$ and $eta$ be its roots.

Fundamental Relations:

The core relations between the roots ($alpha, eta$) and coefficients ($a, b, c$) are:

• 
  Sum of Roots: The sum of the roots is equal to the negative of the coefficient of $x$ divided by the coefficient of $x^2$.
  
  $mathbf{alpha + eta = -frac{b}{a}}$
  


• 
  Product of Roots: The product of the roots is equal to the constant term divided by the coefficient of $x^2$.
  
  $mathbf{alphaeta = frac{c}{a}}$
  

Forming a Quadratic Equation Given its Roots:

If the roots $alpha$ and $eta$ of a quadratic equation are known, the equation can be constructed using the formula:

$mathbf{x^2 - (alpha + eta)x + alphaeta = 0}$

This can also be written as:

$mathbf{x^2 - ( ext{Sum of Roots})x + ( ext{Product of Roots}) = 0}$

Derived Relations (Important for JEE):

Many problems require finding expressions involving powers or differences of roots. These can be calculated using the fundamental sum and product relations:

• 
  Difference of Roots:
  
  $mathbf{|alpha - eta| = frac{sqrt{b^2 - 4ac}}{|a|} = frac{sqrt{D}}{|a|}}$, where $D$ is the discriminant.
  


• 
  Sum of Squares of Roots:
  
  $mathbf{alpha^2 + eta^2 = (alpha + eta)^2 - 2alphaeta}$
  


• 
  Sum of Cubes of Roots:
  
  $mathbf{alpha^3 + eta^3 = (alpha + eta)^3 - 3alphaeta(alpha + eta)}$
  


• 
  Sum of Reciprocals of Roots:
  
  $mathbf{frac{1}{alpha} + frac{1}{eta} = frac{alpha + eta}{alphaeta}}$
  

CBSE vs. JEE Focus:

Aspect	CBSE Board Exams	JEE Main
Core Application	Direct application of sum/product to find values or form equations.	Extensive use in complex problems involving symmetric functions of roots, parameter variations, and transformations of equations.
Derived Relations	Limited to simple transformations like $alpha^2 + eta^2$.	Requires fluency with all common derived relations and algebraic manipulations.

Important Note for JEE:

Always remember that these relations are extremely powerful when dealing with problems where you don't need to find the exact roots, but rather expressions involving them. Mastering these relations saves significant time and effort. Be quick in recognizing symmetric expressions of roots that can be simplified using $alpha+eta$ and $alphaeta$.

Problem-Solving Approach: Relations Between Roots and Coefficients

Understanding the relationship between roots and coefficients is fundamental for solving a wide array of problems in quadratic equations. This section outlines a systematic approach to tackle such problems effectively.

Core Strategy: Vieta's Formulas

The cornerstone of solving problems related to roots and coefficients for a quadratic equation ax² + bx + c = 0 (where a ≠ 0) lies in Vieta's formulas:

• Sum of roots (α + β) = -b/a


• Product of roots (αβ) = c/a

Most problems require you to express a given condition or expression in terms of these two fundamental relations.

Step-by-Step Approach for Problem Solving

1. 
   Identify the Quadratic Equation: Clearly write down the given quadratic equation. If it's not in standard form (ax² + bx + c = 0), rearrange it.
   


2. 
   Apply Vieta's Formulas: Immediately write down the sum and product of roots (α + β and αβ) in terms of the coefficients a, b, and c.
   


3. 
   Analyze the Target Expression/Condition:
   
   
   
   • 
     If you need to find the value of an expression involving α and β, aim to transform it into an expression containing only (α + β) and (αβ).
     
   
   
   • 
     If a condition is given (e.g., one root is double the other), use this condition along with Vieta's formulas to form a system of equations.
     
   
   
   • 
     If you need to form a new quadratic equation whose roots are related to α and β, use Vieta's formulas for the new roots.
     
   
   
   
   


4. 
   Utilize Algebraic Identities: For expressions involving powers of roots (α², β², α³, β³), common identities are crucial:
   
   
   
   • α² + β² = (α + β)² - 2αβ
   
   
   • α³ + β³ = (α + β)³ - 3αβ(α + β)
   
   
   • (α - β)² = (α + β)² - 4αβ
   
   
   • 1/α + 1/β = (α + β) / αβ
   
   
   
   


5. 
   Substitute and Solve: Substitute the values of (α + β) and (αβ) obtained from Vieta's formulas into the transformed expression or system of equations. Solve for the required value or unknown coefficient.
   


6. 
   JEE Tip: Parameter-based Problems: Many JEE problems involve parameters. Your solution will often be an expression in terms of these parameters. Be meticulous with algebraic manipulation. Also, consider edge cases for parameter values (e.g., denominator becoming zero).
   

Illustrative Example

Let α and β be the roots of the quadratic equation x² - 5x + 3 = 0. Find the value of α³/β + β³/α.

Solution Approach:

1. 
   Equation: x² - 5x + 3 = 0. Here, a=1, b=-5, c=3.
   


2. 
   Vieta's Formulas:
   
   
   
   • α + β = -(-5)/1 = 5
   
   
   • αβ = 3/1 = 3
   
   
   
   


3. 
   Target Expression Transformation:
   We need to find α³/β + β³/α.
   
   First, combine the terms:
   α³/β + β³/α = (α⁴ + β⁴) / (αβ)
   
   Now, express α⁴ + β⁴ in terms of (α+β) and (αβ).
   We know α² + β² = (α + β)² - 2αβ = 5² - 2(3) = 25 - 6 = 19.
   
   Then, α⁴ + β⁴ = (α² + β²)² - 2(αβ)²
   = (19)² - 2(3)²
   = 361 - 2(9)
   = 361 - 18 = 343.
   


4. 
   Substitute and Solve:
   α³/β + β³/α = (α⁴ + β⁴) / (αβ) = 343 / 3.
   

This systematic approach ensures that even complex expressions involving roots can be simplified and evaluated efficiently. Master these transformations, and you'll navigate quadratic equation problems with confidence.

The relations between roots and coefficients, often referred to as Vieta's formulas, form a cornerstone of quadratic equations and are frequently tested in JEE Main. Mastery of this section is crucial for solving a wide array of problems, including those that combine concepts from other topics.

Key Formulas & Direct Applications

For a quadratic equation $ax^2 + bx + c = 0$, where $a
eq 0$, let its roots be $alpha$ and $eta$.

• Sum of roots: $alpha + eta = -frac{b}{a}$


• Product of roots: $alphaeta = frac{c}{a}$

An equation with roots $alpha$ and $eta$ can be written as $x^2 - (alpha + eta)x + alphaeta = 0$.

JEE Focus 1: Symmetric Functions of Roots

Many problems require finding expressions involving roots without actually calculating the roots. These are often symmetric functions, meaning their value remains unchanged if $alpha$ and $eta$ are interchanged. Expressing these in terms of $alpha + eta$ and $alphaeta$ is key.

Expression	In terms of Sum and Product
$alpha^2 + eta^2$	$(alpha + eta)^2 - 2alphaeta$
$alpha^3 + eta^3$	$(alpha + eta)^3 - 3alphaeta(alpha + eta)$
$alpha - eta$ (or $eta - alpha$)	$pmsqrt{(alpha + eta)^2 - 4alphaeta}$
$frac{1}{alpha} + frac{1}{eta}$	$frac{alpha + eta}{alphaeta}$

Tip: For higher powers like $alpha^n + eta^n$, a recurrence relation can often be formed. If $alpha, eta$ are roots of $ax^2 + bx + c = 0$, then $aalpha^2 + balpha + c = 0$ and $aeta^2 + beta + c = 0$. Multiplying by $alpha^{n-2}$ and $eta^{n-2}$ respectively and adding gives $a(alpha^n + eta^n) + b(alpha^{n-1} + eta^{n-1}) + c(alpha^{n-2} + eta^{n-2}) = 0$.

JEE Focus 2: Transformation of Equations

This involves finding a new quadratic equation whose roots are related to the roots of a given equation. For example, if $alpha, eta$ are roots of $ax^2+bx+c=0$, find the equation whose roots are $2alpha-1, 2eta-1$.

• Method 1 (Direct): Find the sum and product of the new roots in terms of $alpha+eta$ and $alphaeta$, then form the new equation.


• Method 2 (Substitution): Let $y$ be a root of the new equation. Express $y$ in terms of $x$ (an old root), i.e., $y = f(x)$. Then express $x$ in terms of $y$, i.e., $x = f^{-1}(y)$. Substitute $f^{-1}(y)$ into the original equation $a(f^{-1}(y))^2 + b(f^{-1}(y)) + c = 0$. This will yield the new equation in $y$. This method is generally more efficient for complex transformations.

JEE Focus 3: Common Roots

Problems involving two quadratic equations having common roots.
Let $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$.

• Condition for Exactly One Common Root: If $alpha$ is the common root, then $a_1alpha^2 + b_1alpha + c_1 = 0$ and $a_2alpha^2 + b_2alpha + c_2 = 0$. Solve these simultaneously for $alpha^2$ and $alpha$ using cross-multiplication. Then, $(alpha^2)^2 = (alpha)(alpha)$.


• Condition for Both Roots Common: This occurs if the two quadratic equations are identical or scalar multiples of each other.
  $frac{a_1}{a_2} = frac{b_1}{b_2} = frac{c_1}{c_2}$.

JEE Focus 4: Inter-Topic Connections

This topic frequently combines with:

• Arithmetic Progression (AP), Geometric Progression (GP), Harmonic Progression (HP): Roots being in AP/GP/HP. For example, if roots are in AP, let them be $a-d, a, a+d$.


• Inequalities: Determining parameters for specific root conditions (e.g., real roots, positive roots, roots greater than a certain value). This often links to the discriminant and location of roots.


• Complex Numbers: When coefficients are real, complex roots always occur in conjugate pairs.

Mastering these applications and problem-solving strategies will significantly boost your score in quadratic equations in JEE Main.