The quadratic equation `ax^(2)+bx+c=0` has real roots if:

Options

A. `alt-1,0ltclt1,bgt0`
B. `alt-1,-1ltclt0,0ltblt1`
C. `alt-1,clt0,bgt1`
D. `agt0,bgt0,cgt0`

Answer

To determine when the quadratic equation \( ax^2 + bx + c = 0 \) has real roots, we need to analyze the discriminant of the equation, which is given by:

\[ D = b^2 - 4ac \]

The quadratic equation will have real roots if the discriminant \( D \) is greater than or equal to zero:

\[ D \geq 0 \]

Now, let's evaluate the conditions provided in the question step by step.

### Step 1: Analyze the conditions

1. **Condition 1**: \( a < -1 \), \( c > 0 \)
- Here, \( a \) is negative and \( c \) is positive.
- The discriminant becomes:
\[ D = b^2 - 4ac \]
- Since \( a < -1 \) (negative) and \( c > 0 \) (positive), the term \( -4ac \) becomes positive. Thus, \( D \) will be:
\[ D = b^2 + \text{(positive number)} \]
- Therefore, \( D \) is always positive, and this condition guarantees real roots.

2. **Condition 2**: \( a < -1 \), \( c \) between \( 0 \) and \( 1 \)
- Similar to the first condition, \( a < -1 \) (negative) and \( c \) is positive but less than 1.
- The discriminant is:
\[ D = b^2 - 4ac \]
- Again, \( -4ac \) is positive, making \( D \) a positive number minus a smaller positive number. Thus, we cannot determine if \( D \) is always greater than or equal to zero.

3. **Condition 3**: \( a < 1 \), \( c < 0 \), \( b > 1 \)
- Here, \( a \) is less than 1 (but could be positive or negative), \( c \) is negative, and \( b \) is positive.
- The discriminant becomes:
\[ D = b^2 - 4ac \]
- Since \( c < 0 \), \( -4ac \) becomes positive. Thus, \( D \) is:
\[ D = b^2 + \text{(positive number)} \]
- Again, we cannot determine if \( D \) is always greater than or equal to zero.

4. **Condition 4**: \( a > 0 \), \( b > 0 \), \( c > 0 \)
- All coefficients are positive.
- The discriminant is:
\[ D = b^2 - 4ac \]
- Since all terms are positive, we cannot guarantee that \( D \) is greater than or equal to zero.

### Conclusion

From the analysis:

- **Condition 1** is valid as it guarantees real roots.
- **Conditions 2, 3, and 4** do not guarantee real roots.

Thus, the quadratic equation \( ax^2 + bx + c = 0 \) has real roots if the condition from **Option 1** is satisfied.