Question

If the roots \(x^2 + ax + 9 = 0\) are complex, then

• A. \(a<6\)
• B. \(a<-6\)
• C. \(|a|<6\)
• D. \(|a|>6\)

Hint:

For complex roots: discriminant must be negative.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
For a quadratic equation \( Ax^{2} + Bx + C = 0 \), the roots are complex (non-real) if the discriminant (\( D \)) is less than zero.
: Key Formula or Approach:
Discriminant \( D = B^{2} - 4AC \).
For complex roots, \( D<0 \).
Step 2: Detailed Explanation:
In the given equation \( x^{2} + ax + 9 = 0 \), we have:
\( A = 1, B = a, C = 9 \).
Calculating the discriminant:
\[ D = a^{2} - 4(1)(9) = a^{2} - 36 \]
Since the roots are complex:
\[ D<0 \implies a^{2} - 36<0 \]
\[ a^{2}<36 \]
Taking the square root on both sides leads to the absolute value inequality:
\[ |a|<6 \]
This means \( -6<a<6 \).
Step 3: Final Answer:
The condition for complex roots is \( |a|<6 \).