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 step by step, we start with the given equations and conditions.

### Step 1: Understand the Given Information
We are given:
1. The equation \(4a + 2b + c = 0\).
2. The condition \(ab > 0\).
3. The quadratic equation \(ax^2 + bx + c = 0\).

### Step 2: Substitute Values into the Quadratic Equation
We need to analyze the quadratic equation \(ax^2 + bx + c = 0\). We can substitute \(x = 2\) into the equation:

\[ a(2^2) + b(2) + c = 0 \]

This simplifies to:

\[ 4a + 2b + c = 0 \]

### Step 3: Use the Given Condition
Since we know from the problem statement that \(4a + 2b + c = 0\), we can conclude that \(x = 2\) is a root of the quadratic equation.

### Step 4: Analyze the Nature of the Roots
Since \(x = 2\) is a root, we can factor the quadratic equation as follows:

\[ a(x - 2)(x - r) = 0 \]

where \(r\) is the other root.

### Step 5: Determine the Nature of the Other Root
To determine the nature of the roots, we need to consider the condition \(ab > 0\). This condition implies that both \(a\) and \(b\) are either both positive or both negative.

1. If \(a > 0\) and \(b > 0\), then the parabola opens upwards, and since one root is \(2\), the other root \(r\) must also be real and greater than \(2\).
2. If \(a < 0\) and \(b < 0\), then the parabola opens downwards, and since one root is \(2\), the other root \(r\) must also be real and less than \(2\).

### Step 6: Conclusion
In both cases, we find that the quadratic equation has two real roots: one root is \(2\) and the other root \(r\) is real as well. Therefore, the equation \(ax^2 + bx + c = 0\) has two real roots.

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

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