Question

If the difference between the roots of $x^2 + ax + b = 0$ and that of $x^2 + bx + a = 0$ is same and $a \ne b$, then

• A. $a - b - 4 = 0$
• B. $a - b + 4 = 0$
• C. $a + b + 4 = 0$
• D. $a + b - 4 = 0$

Hint:

Use root difference formulas and square both sides carefully when equating root-based expressions.

Answer 1

The Correct Option is B

For $x^2 + ax + b = 0$, difference of roots = $\sqrt{a^2 - 4b}$
For $x^2 + bx + a = 0$, difference = $\sqrt{b^2 - 4a}$
Given: $\sqrt{a^2 - 4b} = \sqrt{b^2 - 4a}$
Square both sides: $a^2 - 4b = b^2 - 4a$
Rearranged: $a^2 - b^2 = 4b - 4a \Rightarrow (a - b)(a + b) = 4(b - a)$
$\Rightarrow -(a - b)(a + b) = 4(a - b) \Rightarrow -(a + b) = 4$
$\Rightarrow a + b = -4 \Rightarrow a - b = -4 - 2b \Rightarrow a - b + 4 = 0$