Let a, b and c be real numbers such that `4a + 2b + c = 0` and `ab gt 0.` Then the equation ax^(2) + bx + c = 0` has

Options

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

Explanation

To solve the problem, we need to analyze the given conditions and the quadratic equation \( ax^2 + bx + c = 0 \).

### Step 1: Understand the given conditions
We have two conditions:
1. \( 4a + 2b + c = 0 \)
2. \( ab > 0 \)

From the second condition, \( ab > 0 \), we can infer that both \( a \) and \( b \) must either be positive or both must be negative. This means that \( a \) and \( b \) cannot be zero.

### Step 2: Substitute the values into the quadratic equation
The quadratic equation is given as:
\[ ax^2 + bx + c = 0 \]
We need to check the nature of the roots of this equation.

### Step 3: Calculate \( f(2) \)
We can evaluate the quadratic function at \( x = 2 \):
\[ f(2) = a(2^2) + b(2) + c = 4a + 2b + c \]
From the first condition, we know that:
\[ 4a + 2b + c = 0 \]
Thus, we have:
\[ f(2) = 0 \]
This shows that \( x = 2 \) is a root of the equation \( ax^2 + bx + c = 0 \).

### Step 4: Determine the nature of the roots
Since we have found that \( x = 2 \) is a root, we can conclude that the quadratic equation has at least one real root.

### Step 5: Analyze the implications of \( ab > 0 \)
Given that \( ab > 0 \), we know that both \( a \) and \( b \) are either positive or negative. This means the parabola opens upwards if \( a > 0 \) or downwards if \( a < 0 \).

### Step 6: Conclusion
Since we have established that the quadratic has a real root (specifically \( x = 2 \)), we can conclude that the equation \( ax^2 + bx + c = 0 \) has real roots.

Thus, the answer is that the equation has real roots.

### Final Answer
The equation \( ax^2 + bx + c = 0 \) has real roots.

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