If `a,b,c in R and a+b+c=0`, then the quadratic equation `4ax^(2)+3bx +2c=0` has

Options

A. one positive and one negative root
B. imaginary roots
C. real roots
D. none of these

Answer

To solve the problem, we need to analyze the quadratic equation given by \(4ax^2 + 3bx + 2c = 0\) under the condition \(a + b + c = 0\).

### Step-by-step Solution:

1. **Identify the coefficients**:
- The coefficients of the quadratic equation are:
- \(A = 4a\)
- \(B = 3b\)
- \(C = 2c\)

2. **Calculate the discriminant**:
- The discriminant \(D\) of a quadratic equation \(Ax^2 + Bx + C = 0\) is given by:
\[ D = B^2 - 4AC \]
- Substituting the values of \(A\), \(B\), and \(C\):
\[ D = (3b)^2 - 4(4a)(2c) = 9b^2 - 32ac \]

3. **Substitute \(b\)**:
- From the condition \(a + b + c = 0\), we can express \(b\) in terms of \(a\) and \(c\):
\[ b = -a - c \]
- Substitute this into the discriminant:
\[ D = 9(-a - c)^2 - 32ac \]

4. **Expand the expression**:
- Expanding \(D\):
\[ D = 9(a^2 + 2ac + c^2) - 32ac \]
\[ D = 9a^2 + 18ac + 9c^2 - 32ac \]
\[ D = 9a^2 - 14ac + 9c^2 \]

5. **Analyze the discriminant**:
- The discriminant \(D = 9a^2 - 14ac + 9c^2\) can be rewritten as:
\[ D = 9(a^2 + c^2) - 14ac \]
- This expression is a quadratic in terms of \(a\) and \(c\).

6. **Determine the nature of the roots**:
- The expression \(D\) is a quadratic function in \(a\) and \(c\). Since \(9(a^2 + c^2)\) is always non-negative and \(14ac\) is a product of two real numbers, we can conclude:
- The discriminant \(D\) can be zero or positive, depending on the values of \(a\) and \(c\).
- If \(D > 0\), the roots are real and distinct.
- If \(D = 0\), the roots are real and equal.

### Conclusion:
Given that \(D\) can be zero or positive, we conclude that the quadratic equation \(4ax^2 + 3bx + 2c = 0\) has real roots.