The equation `ax^(2) +bx+c=0` where a,b,c are real numbers connected by the relation `4a +2b+c=0 and ab gt 0` has

Options

A. real roots
B. complex roots
C. exactly one root
D. none of these

Answer

To solve the quadratic equation \( ax^2 + bx + c = 0 \) given the conditions \( 4a + 2b + c = 0 \) and \( ab > 0 \), we will analyze the roots of the equation step by step.

### Step 1: Understand the given conditions
We have the quadratic equation:
\[ ax^2 + bx + c = 0 \]
with the conditions:
1. \( 4a + 2b + c = 0 \)
2. \( ab > 0 \)

### Step 2: Substitute the value of \( c \)
From the first condition, we can express \( c \) in terms of \( a \) and \( b \):
\[ c = -4a - 2b \]

### Step 3: Rewrite the quadratic equation
Substituting \( c \) back into the quadratic equation gives:
\[ ax^2 + bx - (4a + 2b) = 0 \]
This simplifies to:
\[ ax^2 + bx - 4a - 2b = 0 \]

### Step 4: Calculate the discriminant
The discriminant \( D \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
Substituting our expression for \( c \):
\[ D = b^2 - 4a(-4a - 2b) \]
\[ D = b^2 + 16a^2 + 8ab \]

### Step 5: Analyze the discriminant
Since \( ab > 0 \), both \( a \) and \( b \) must have the same sign (both positive or both negative). This implies:
- \( 16a^2 \) is always positive.
- \( 8ab \) is also positive because \( ab > 0 \).

Thus, we can conclude that:
\[ D = b^2 + 16a^2 + 8ab > 0 \]
This means the discriminant is positive.

### Step 6: Determine the nature of the roots
Since the discriminant \( D > 0 \), the quadratic equation has two distinct real roots.

### Conclusion
The equation \( ax^2 + bx + c = 0 \) has two distinct real roots.

### Final Answer
The correct option is that the equation has **two distinct real roots**.

---