Difference between the corresponding roots of x² + ax + b = 0 and x² + bx + a = 0 is same and a ≠ b,

Q: Wait, why do we reject the case where $a - b = 0$?

That is a fantastic observation! In the final step, we arrive at $(a - b)(a + b + 4) = 0$. Mathematically, this implies either $a - b = 0$ or $a + b + 4 = 0$. However, the problem statement explicitly provides the condition $a \neq b$. This means $a - b$ cannot be zero. Therefore, the only remaining possibility is that $a + b + 4 = 0$. It is crucial to always check the constraints given in the problem!

Q: What if the roots are imaginary? Does the formula for the difference of roots still hold?

Great question! The formula $|\alpha - \beta| = \frac{\sqrt{D}}{|A|}$ is derived from the quadratic formula $\frac{-B \pm \sqrt{D}}{2A}$. Even if $D < 0$, the difference $\alpha - \beta$ is still $\frac{\sqrt{D}}{A}$ (or its negative). The absolute value handles the sign, and the logic remains consistent. In JEE problems, unless specified otherwise, we treat the algebraic relationship as valid regardless of the nature of the roots.

Q: Why can't we just find the roots using the quadratic formula and subtract them?

You absolutely could! However, calculating $\frac{-a + \sqrt{a^2 - 4b}}{2} - \frac{-a - \sqrt{a^2 - 4b}}{2}$ is algebraically tedious and prone to sign errors. Using the discriminant property $|\alpha - \beta| = \frac{\sqrt{D}}{|A|}$ is a much more elegant and 'JEE-smart' way to handle the problem. It saves time and reduces the chance of making a silly mistake.

Question

Difference between the corresponding roots of $x^2 + ax + b = 0$ and $x^2 + bx + a = 0$ is same and $a 
eq b$, then

Options:

(A) $a + b + 4 = 0$

(B) $a + b - 4 = 0$

(C) $a - b - 4 = 0$

(D) $a - b + 4 = 0$

Sigma Percentile · Accepted Answer

### The Geometry of Roots: A Journey into Parabolas

Imagine you are standing in front of two beautiful, sweeping parabolas on a coordinate plane. The first is defined by $x^2 + ax + b = 0$, and the second is defined by $x^2 + bx + a = 0$.

These aren't just equations; they are geometric entities. The roots of these equations are the points where these parabolas intersect the $x$-axis.

The problem asks us to consider the distance between these roots, known as the "difference of roots." When we are told this distance is the same for both, we are being given a powerful geometric constraint.

### The Algebraic Toolkit

To solve this, we don't need to find the roots individually. Instead, we use the discriminant.

For any quadratic equation $Ax^2 + Bx + C = 0$, the difference between the roots $\alpha$ and $\beta$ is given by the elegant relation:

$$|\alpha - \beta| = \frac{\sqrt{D}}{|A|}$$

where $D = B^2 - 4AC$. This formula is our bridge, connecting the coefficients of the equation directly to the distance between the roots.

For our first equation, $x^2 + ax + b = 0$, the leading coefficient $A$ is $1$, and the discriminant $D_1$ is $a^2 - 4b$. Thus, the difference is $\sqrt{a^2 - 4b}$.

For our second equation, $x^2 + bx + a = 0$, the discriminant $D_2$ is $b^2 - 4a$, and the difference is $\sqrt{b^2 - 4a}$.

### The Dance of Manipulation

Now, we equate these two expressions because the problem states they are equal:

$$\sqrt{a^2 - 4b} = \sqrt{b^2 - 4a}$$

Squaring both sides to eliminate the radicals, we obtain:

$$a^2 - 4b = b^2 - 4a$$

Now, we bring everything to one side to reveal the underlying structure:

$$a^2 - b^2 + 4a - 4b = 0$$

We can group the terms as follows:

$$(a^2 - b^2) + 4(a - b) = 0$$

Using the difference of squares identity, $a^2 - b^2 = (a - b)(a + b)$, we transform the equation into:

$$(a - b)(a + b) + 4(a - b) = 0$$

### The Final Revelation

Factoring out the common term $(a - b)$, we get:

$$(a - b)(a + b + 4) = 0$$

We have a product of two terms equal to zero. Given the constraint $a 
eq b$, the term $(a - b)$ cannot be zero.

Therefore, the only logical conclusion is that the second factor must be zero:

$$a + b + 4 = 0$$

We have arrived at the solution, not by brute force, but by understanding the elegant relationship between coefficients and roots. The final result is **$a + b = -4$**.

Animated Solution for Mathematics - Quadratic Equations: Difference between the corresponding roots of $x^{2} + a x + b = 0$ and $x^{2} + b x + a = 0$ is same and $a \neq = b$ , then

Select Answer:

Visualized Solution

Visualizing the Equations

The Given Constraint

Formula for Difference of Roots

Applying the Formula

Equating the Differences

Squaring Both Sides

Rearranging the Terms

Factorizing the Expression

Extracting the Common Factor

Final Conclusion

The Sigma Insight: Relation Between Roots and Coefficients

The Geometry of Roots

The Algebraic Toolkit

The Dance of Manipulation

The Final Revelation

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Animated Solution for Mathematics - Quadratic Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Difference between the corresponding roots of x2+ax+b=0 and x2+bx+a=0 is same and a=b, then