Consider a quadratic equaiton `az^(2) + bz + c=0`, where a,b,c are complex number.
The condition that the equaiton has one purely real roots is

Options

A. `(abarb+barab)(b barc-barbc)=(c bara+barca)^(2)`
B. `(abarb-barab)(b barc+barbc)=(cbara+barca)^(2)`
C. `(abarb-barab)(barbc-barbc)=(cbara-barca)^(2)`
D. `(abarb-barab)(b barc-barbc)=(cbara+barca)^(2)`

Answer

To determine the condition under which the quadratic equation \( az^2 + bz + c = 0 \) has one purely real root, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Roots**:
A purely real root \( z_1 \) implies that \( z_1 \) is equal to its conjugate, i.e., \( z_1 = \overline{z_1} \).

2. **Substituting the Root into the Equation**:
Since \( z_1 \) is a root of the equation, we substitute it into the quadratic equation:
\[ az_1^2 + bz_1 + c = 0 \]

3. **Taking the Conjugate of the Equation**:
Taking the conjugate of the entire equation gives:
\[ \overline{az_1^2 + bz_1 + c} = 0 \]
Since \( z_1 \) is real, we have:
\[ \overline{a}z_1^2 + \overline{b}z_1 + \overline{c} = 0 \]

4. **Equating the Two Equations**:
Now we have two equations:
\[ az_1^2 + bz_1 + c = 0 \quad \text{(1)} \]
\[ \overline{a}z_1^2 + \overline{b}z_1 + \overline{c} = 0 \quad \text{(2)} \]
For the quadratic to have one purely real root, these two equations must be equivalent.

5. **Setting Up the Condition**:
For the two equations to have the same root, the coefficients must satisfy certain conditions. Specifically, we can derive the condition by eliminating \( z_1 \) from the equations.

6. **Using the Discriminant**:
The condition for the quadratic equation to have a double root (which is a purely real root in this case) is that the discriminant must be zero:
\[ D = b^2 - 4ac = 0 \]
However, since \( a, b, c \) are complex, we also need to consider their conjugates.

7. **Final Condition**:
The condition for the quadratic equation \( az^2 + bz + c = 0 \) to have one purely real root is:
\[ b^2 - 4ac = 0 \quad \text{and} \quad a, b, c \text{ must satisfy the conjugate conditions.} \]