Let $f(x) = x^2 - ax + b , a $ is odd positive integer and the roots of the equation $f(x) = 0$ are two distinct prime numbers. If $ a+ b =35$, then the value of $f(10)=$

Since the roots of x^2 - ax + b = 0 are two distinct primes, let them be p and q. By Vieta's formulas, a = p + q and b = pq. Since a is an odd integer, one of the prime roots must be 2 and the other must be an odd prime. Substituting a + b = 35 gives (2 + q) + 2q = 35, so 3q = 33 and q = 11. Therefore, the roots are 2 and 11, making a = 13 and b = 22. The function evaluates to f(10) = 10^2 - 13(10) + 22 = 100 - 130 + 22 = -8.

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