Question

If \( (3 + i) \) is a root of \( x^2 + ax + b = 0 \), then \( a = {?} \)

• A. \( 3 \)
• B. \( \)
• C. \( 6 \)
• D. \( -6 \)

Hint:

For a quadratic equation with real coefficients, if a complex number is a root, its conjugate is also a root. Use the sum and product of roots to find the coefficients.

Answer 1

The Correct Option is D

We are given that \( (3 + i) \) is a root of the quadratic equation: \[ x^2 + ax + b = 0. \] Since the coefficients of the quadratic equation are real numbers, the complex roots of the equation occur in conjugate pairs. Thus, the other root of the equation must be \( (3 - i) \).
Step 1: The sum and product of the roots of a quadratic equation \( x^2 + ax + b = 0 \) are related to the coefficients as follows: - The sum of the roots is \( -a \), - The product of the roots is \( b \). Let the roots of the equation be \( 3 + i \) and \( 3 - i \). We can now calculate the sum and product of the roots: \[ \text{Sum of the roots} = (3 + i) + (3 - i) = 6, \] \[ \text{Product of the roots} = (3 + i)(3 - i) = 3^2 - i^2 = 9 + 1 = 10. \] 
Step 2: From the sum of the roots, we know that: \[ -a = 6 \quad \Rightarrow \quad a = -6. \] Thus, the value of \( a \) is \( -6 \).