Quadratics with primes!

Algebra Level pending

If the zeroes of the quadratic polynomial \(f(x)=x^{2}+bx+c\) are prime numbers such that the \(x\)-coordinate of vertex of curve of \(y=x^2+bx+c\) is not an integer. And if \(f(7)>0\) and \(4(5+b)+c<4\), then find the values of \(b\) and \(c\).

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