Question

Let \( f(x) = x^{2} + ax + b \), where \( a, b \in \mathbb{R} \). If \( f(x)=0 \) has all its roots imaginary, then the roots of \( f(x) + f'(x) + f''(x) = 0 \) are

• A. real and distinct
• B. imaginary
• C. equal
• D. rational and equal

Hint:

If a quadratic $f(x)$ has no real roots, then the sum of $f(x)$ and its derivatives also has no real roots.

Answer 1

The Correct Option is B

Step 1: Given Condition
For $f(x) = x^2 + ax + b$ to have imaginary roots, the discriminant $D<0$, so $a^2 - 4b<0$.
Step 2: Find Derivatives
$f'(x) = 2x + a$ and $f''(x) = 2$.
Step 3: New Equation
$f(x) + f'(x) + f''(x) = x^2 + ax + b + 2x + a + 2 = x^2 + (a+2)x + (a+b+2) = 0$.
Step 4: Check New Discriminant
$D' = (a+2)^2 - 4(a+b+2) = a^2 + 4a + 4 - 4a - 4b - 8 = a^2 - 4b - 4$. Since $a^2 - 4b<0$, then $a^2 - 4b - 4$ must also be less than zero. Thus, roots are imaginary.
Final Answer: (b)