If a,b,c belong to `R,ane0,` and the quadratic equation `ax^2+bx+c=0` has no real roots. Then , which one of the following is correct?

Options

A. `(a+b+c)agt0`
B. `(a+b+c)clt0`
C. `(a+b+c)c=0`
D. `(a+b+c)cle0`

Answer

To solve the problem, we need to analyze the conditions under which the quadratic equation \( ax^2 + bx + c = 0 \) has no real roots. The key point is that a quadratic equation has no real roots when its discriminant is less than zero.

### Step-by-Step Solution:

1. **Understanding the Condition for No Real Roots**:
The quadratic equation \( ax^2 + bx + c = 0 \) has no real roots if the discriminant \( D \) is less than zero. The discriminant is given by:
\[ D = b^2 - 4ac \]
Therefore, for no real roots:
\[ b^2 - 4ac < 0 \quad \text{(1)} \]

2. **Analyzing the Sign of \( a \)**:
Since \( a \neq 0 \), we have two cases to consider:
- Case 1: \( a > 0 \)
- Case 2: \( a < 0 \)

3. **Case 1: \( a > 0 \)**:
If \( a > 0 \), the parabola opens upwards. For the parabola to have no real roots, it must lie entirely above the x-axis. This means that the value of the quadratic function at any point must be greater than zero. Particularly, we can evaluate the function at \( x = 1 \):
\[ f(1) = a(1^2) + b(1) + c = a + b + c > 0 \quad \text{(2)} \]

4. **Case 2: \( a < 0 \)**:
If \( a < 0 \), the parabola opens downwards. For the parabola to have no real roots, it must lie entirely below the x-axis. Thus, the value of the quadratic function at any point must be less than zero. Evaluating at \( x = 1 \):
\[ f(1) = a(1^2) + b(1) + c = a + b + c < 0 \quad \text{(3)} \]

5. **Combining the Results**:
- From (2), if \( a > 0 \), then \( a + b + c > 0 \).
- From (3), if \( a < 0 \), then \( a + b + c < 0 \).

6. **Conclusion**:
In both cases, we can conclude:
- If \( a > 0 \), then \( a + b + c > 0 \).
- If \( a < 0 \), then \( a + b + c < 0 \).

Therefore, we can multiply \( a \) (which is either positive or negative) with \( a + b + c \) (which is also positive or negative respectively) to conclude that:
\[ a(a + b + c) > 0 \quad \text{(4)} \]

### Final Answer:
The correct option is:
\[ \text{a + b + c} \cdot a > 0 \]