Statement-1 : The equation `x^(2)-2009x+2008 =0` has rational roots .
and
Statement -2 : The quadratic equation `ax^(2)+bx+c=0` has rational roots iff `b^(2)-4ac` is a prefect square.

Options

A. Statement -1 is True, Statement -2 is True, Statement -2 is a correct explanation for statement -12
B. Statement -1 is True, Statement -2 is True , Statement -2 is NOT a correct explanation for Statement -12
C. Statement -1 is True, Statement -2 is False
D. Statement -1 is Flase, Statement -2 is True

Answer

To determine whether the statements are true, we can analyze them step by step.

### Step 1: Analyze Statement 1
We need to check if the quadratic equation \( x^2 - 2009x + 2008 = 0 \) has rational roots.

### Step 2: Use the Quadratic Formula
The roots of a quadratic equation \( ax^2 + bx + c = 0 \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our case, \( a = 1 \), \( b = -2009 \), and \( c = 2008 \).

### Step 3: Calculate the Discriminant
We need to calculate the discriminant \( D = b^2 - 4ac \):
\[ D = (-2009)^2 - 4 \cdot 1 \cdot 2008 \]
Calculating \( (-2009)^2 \):
\[ D = 2009^2 - 4 \cdot 2008 \]
Calculating \( 2009^2 \):
\[ 2009^2 = 4036081 \]
Calculating \( 4 \cdot 2008 \):
\[ 4 \cdot 2008 = 8032 \]
Now substituting back:
\[ D = 4036081 - 8032 = 4038049 \]

### Step 4: Check if the Discriminant is a Perfect Square
Next, we need to check if \( 4038049 \) is a perfect square:
\[ \sqrt{4038049} = 2009 \]
Since \( 2009^2 = 4038049 \), the discriminant is indeed a perfect square.

### Step 5: Conclusion for Statement 1
Since the discriminant is a perfect square, the roots of the equation are rational. Therefore, Statement 1 is true.

### Step 6: Analyze Statement 2
Statement 2 states that a quadratic equation \( ax^2 + bx + c = 0 \) has rational roots if and only if \( b^2 - 4ac \) is a perfect square. This is a well-known result in algebra.

### Step 7: Conclusion for Statement 2
Since this statement is a standard result in quadratic equations, Statement 2 is also true.

### Final Conclusion
Both statements are true, and Statement 1 is a correct explanation of Statement 2.