Let $f\left(x\right)=x^2-ax+b$, and $'a'$ be an odd positive integer and the roots of the equation $:f\left(x\right)=0$ are two distinct prime numbers. If $a+b=35$, then

Let the two distinct prime roots of x^2 - ax + b = 0 be p and q, giving a = p + q and b = pq. Since a is an odd positive integer and p and q are distinct primes, one prime must be 2 and the other must be odd. Testing prime pairs gives p = 2 and q = 15 as invalid, so testing a + b = 35 yields 19 when p = 2 and q = 17, making the sum of the roots a = 19 and the product b = 34. Re-evaluating the condition a + b = 35 with valid primes gives the correct pair as 2 and 15 being invalid, so we find the actual roots satisfying the sum are distinct primes summing to a specific odd integer. The sum of the roots is 17.

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