Statement -1 : The quadratic equation `ax^(2)+bx+c=0` has real roots if ` (a+c)^(2) gt b^(2) , AA , a,b,c in R` .
and
Statement -2 : The quadratic equation `ax^(2) +bx+c=0` has real roots if ` b^(2) -4ac ge 0`

Options

A. Statement -1 is True, Statement -2 is True, Statement -2 is a correct explanation for statement -14
B. Statement -1 is True, Statement -2 is True , Statement -2 is NOT a correct explanation for Statement -14
C. Statement -1 is True, Statement -2 is False
D. Statement -1 is Flase, Statement -2 is True

Answer

To solve the problem, we need to analyze both statements regarding the conditions under which a quadratic equation has real roots.

### Step-by-Step Solution:

1. **Understanding the Quadratic Equation**:
The general form of a quadratic equation is given by:
\[ ax^2 + bx + c = 0 \]
where \(a\), \(b\), and \(c\) are real numbers.

2. **Condition for Real Roots**:
A quadratic equation has real roots if its discriminant \(D\) is non-negative. The discriminant is given by:
\[ D = b^2 - 4ac \]
Therefore, for the quadratic equation to have real roots, we need:
\[ b^2 - 4ac \geq 0 \]

3. **Analyzing Statement 1**:
The first statement claims that the quadratic equation has real roots if:
\[ (a + c)^2 > b^2 \]
We can expand this:
\[ a^2 + 2ac + c^2 > b^2 \]
This does not necessarily imply that \(b^2 - 4ac \geq 0\). In fact, it is possible for \((a + c)^2 > b^2\) to hold while \(b^2 - 4ac < 0\). Therefore, Statement 1 is **incorrect**.

4. **Analyzing Statement 2**:
The second statement claims that the quadratic equation has real roots if:
\[ b^2 - 4ac \geq 0 \]
This is the standard condition for a quadratic equation to have real roots. Thus, Statement 2 is **correct**.

5. **Conclusion**:
- Statement 1 is false.
- Statement 2 is true.

### Final Answer:
- Statement 1: False
- Statement 2: True