Question

If \(a, b, c\) are distinct odd natural numbers, then the number of rational roots of the equation \(ax^2 + bx + c = 0\) is:

• A. must be 0
• B. must be 1
• C. must be 2
• D. cannot be determined from the given data

Hint:

To check for rational roots, compute the discriminant and determine if it is a perfect square. If not, there are no rational roots.

Answer 1

The Correct Option is A

Step 1: The quadratic equation is:

\[ ax^2 + bx + c = 0 \]

where \( a, b, c \) are distinct odd natural numbers.

Step 2: The discriminant \( \Delta \) of the quadratic equation is:

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

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

Step 3: Since \( a, b, c \) are distinct odd natural numbers, \( b^2 \) is odd, and \( 4ac \) is also odd (since \( a \) and \( c \) are odd). Thus, \( b^2 - 4ac \) 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.