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 $-6$.