Let `alpha and beta` be real numbers and z be a complex number. If `z^(2)+alphaz+beta=0` has two distinct non-real roots with Re(z)=1, then it is necessary that

Options

A. `beta in (-1,0)`
B. `abs(beta)=1`
C. `beta in (1,infty)`
D. `beta in (0,1)`

Answer

To solve the problem step by step, we start with the given quadratic equation:

### Step 1: Write down the quadratic equation
The quadratic equation is given by:
\[ z^2 + \alpha z + \beta = 0 \]

### Step 2: Use the quadratic formula
The roots of the quadratic equation can be found using the quadratic formula:
\[ z = \frac{-\alpha \pm \sqrt{\alpha^2 - 4\beta}}{2} \]

### Step 3: Analyze the real part of the roots
According to the problem, the real part of the roots is given to be 1. Therefore, we can set the real part of the roots equal to 1:
\[ \frac{-\alpha}{2} = 1 \]

### Step 4: Solve for \(\alpha\)
From the equation above, we can solve for \(\alpha\):
\[ -\alpha = 2 \implies \alpha = -2 \]

### Step 5: Determine the condition for distinct non-real roots
For the roots to be distinct and non-real, the discriminant must be less than zero:
\[ \alpha^2 - 4\beta < 0 \]

### Step 6: Substitute \(\alpha\) into the discriminant condition
Substituting \(\alpha = -2\) into the discriminant condition:
\[ (-2)^2 - 4\beta < 0 \]
\[ 4 - 4\beta < 0 \]

### Step 7: Solve the inequality
Now, we simplify the inequality:
\[ 4 < 4\beta \]
\[ 1 < \beta \]
This implies:
\[ \beta > 1 \]

### Conclusion
Thus, the necessary condition is that \(\beta\) must be greater than 1.

### Final Answer
\(\beta \in (1, \infty)\)

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