Question

If the difference between the roots of \[ x^2+ax+b=0 \] and that of the roots of \[ x^2+bx+a=0 \] is same and \(a\neq b\), then

• A. \(a-b-4=0\)
• B. \(a-b+4=0\)
• C. \(a+b+4=0\)
• D. \(a+b-4=0\)

Hint:

For quadratic equations, the difference between roots depends on the discriminant: \[ \text{Difference of roots}=\sqrt{D} \] when the coefficient of \(x^2\) is \(1\).

Answer 1

The Correct Option is C

Step 1: Recall the spread of roots.
For a monic quadratic $x^2+px+q=0$ the difference of roots has magnitude $\sqrt{p^2-4q}$ (the square root of the discriminant).
Step 2: Difference for the first equation.
For $x^2+ax+b=0$ the difference is $\sqrt{a^2-4b}$.
Step 3: Difference for the second equation.
For $x^2+bx+a=0$ the difference is $\sqrt{b^2-4a}$.
Step 4: Set the differences equal.
Equality gives $a^2-4b=b^2-4a$.
Step 5: Rearrange.
$a^2-b^2=4b-4a$, so $(a-b)(a+b)=-4(a-b)$.
Step 6: Cancel and conclude.
Since $a\neq b$, divide by $(a-b)$ to get $a+b=-4$, i.e. $a+b+4=0$, which is option (3).
\[ \boxed{a+b+4=0} \]