Let ` a,b,c in R and a ne 0` be such that ` (a + c)^(2) lt b^(2) ` ,then the quadratic equation ` ax^(2) + bx +c =0` has

Options

A. Imaginary roots
B. Real roots
C. Exactly one real root lying in the interval (-1,1)
D. Exactly two roots in (-1,1)

Answer

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

### Step-by-Step Solution:

1. **Understanding the Given Inequality**:
We are given that \( (a + c)^2 < b^2 \). This means that the square of the sum \( a + c \) is less than the square of \( b \).

2. **Expanding the Inequality**:
From the inequality \( (a + c)^2 < b^2 \), we can expand it:
\[ a^2 + 2ac + c^2 < b^2 \]

3. **Rearranging the Inequality**:
We can rearrange the inequality to isolate \( b^2 \):
\[ b^2 - (a^2 + 2ac + c^2) > 0 \]
This can be rewritten as:
\[ b^2 - 4ac > 0 \]

4. **Identifying the Discriminant**:
The discriminant \( D \) of the quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[ D = b^2 - 4ac \]
From our rearrangement, we have \( D > 0 \).

5. **Conclusion on the Nature of Roots**:
Since the discriminant \( D \) is greater than 0, it indicates that the quadratic equation has two distinct real roots.

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

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