Quadratic Equation

Step 1: The quadratic equation is:

where are distinct odd natural numbers.

Step 2: The discriminant of the quadratic equation is:

For the quadratic to have rational roots, the discriminant must be a perfect square.

Step 3: Since are distinct odd natural numbers, is odd, and is also odd (since and are odd). Thus, is even.

Step 4: However, the difference of an odd number and an even number is always odd, so the discriminant cannot be a perfect square.

Step 5: Therefore, the equation has no rational roots.

Step 1: We are given the equations and , and we need to solve for the equation .

Step 2: Subtract from :

Simplifying:

Step 3: The resulting equation is:

Step 4: Factor out :

Step 5: The solutions to this equation are or . The second equation gives:

So, the equation has at least two real roots: and .

Step 1: The given function is . We are required to find the relations between the terms for the equation , where and are the first and second derivatives of .

Step 2: First, compute the first and second derivatives of . Use the quotient rule for differentiation:

Then, compute the second derivative .

Step 3: Now, substitute and into the equation .

Step 4: After solving for , we find that the correct values are:

1. The sum and product of the roots of the quadratic equation are:

2. Given and :

• , implying since .
• , implying because .

3. Consider :

• Since and , does not hold in general.

4. Consider :

• Substitute the values of and to test:
• , as .

Thus, the correct answers are (A) and (C).