If `alpha and beta` are non-real, then condition for `x^(2) + alpha x + beta = 0` to have real roots, is

Options

A. `(alpha-bar(alpha))(beta-bar(beta))=(alpha bar(beta)-bar(alpha)beta)^(2)`
B. `(bar(alpha)-alpha)(alphabar(beta)-bar(alpha)beta)=(beta-bar(beta))^(2)`
C. `(beta-bar(beta))(alpha bar(beta)-bar(alpha)beta)=(bar(alpha)-alpha)^(2)`
D. none of these

Explanation

To determine the condition for the quadratic equation \( x^2 + \alpha x + \beta = 0 \) to have real roots, given that \( \alpha \) and \( \beta \) are non-real, we can follow these steps:

### Step 1: Understand the Quadratic Equation
The standard form of a quadratic equation is \( ax^2 + bx + c = 0 \). In our case, \( a = 1 \), \( b = \alpha \), and \( c = \beta \).

### Step 2: Use the Discriminant
For a quadratic equation to have real roots, the discriminant must be non-negative. The discriminant \( D \) is given by:
\[ D = b^2 - 4ac \]
Substituting our values:
\[ D = \alpha^2 - 4 \cdot 1 \cdot \beta = \alpha^2 - 4\beta \]

### Step 3: Set the Discriminant Condition
Since we want the roots to be real, we need:
\[ D \geq 0 \implies \alpha^2 - 4\beta \geq 0 \]

### Step 4: Analyze the Condition
Given that \( \alpha \) and \( \beta \) are non-real, we can express them in terms of their real and imaginary parts. Let:
\[ \alpha = a + bi \quad \text{and} \quad \beta = c + di \]
where \( a, b, c, d \) are real numbers and \( b \neq 0 \), \( d \neq 0 \).

### Step 5: Substitute and Simplify
Substituting \( \alpha \) into the discriminant condition:
\[ (a + bi)^2 - 4(c + di) \geq 0 \]
Calculating \( (a + bi)^2 \):
\[ (a + bi)^2 = a^2 - b^2 + 2abi \]
Thus, the discriminant becomes:
\[ (a^2 - b^2 + 2abi) - 4(c + di) = (a^2 - b^2 - 4c) + (2ab - 4d)i \]

### Step 6: Set Real and Imaginary Parts
For the discriminant to be non-negative, the imaginary part must equal zero:
\[ 2ab - 4d = 0 \quad \text{(1)} \]
And the real part must be non-negative:
\[ a^2 - b^2 - 4c \geq 0 \quad \text{(2)} \]

### Step 7: Conclusion
Thus, the conditions for the quadratic equation \( x^2 + \alpha x + \beta = 0 \) to have real roots, given that \( \alpha \) and \( \beta \) are non-real, are:
1. \( 2ab - 4d = 0 \)
2. \( a^2 - b^2 - 4c \geq 0 \)