\(314_{10} - 256_{7} = 340_x\),

Convert \(256_{7}\) and \(34_{x} to base 10, such that:

\(314 - 139 = 3x^2 + 4x\)

=> \(3x^2 + 4x - 175 = 0\) (quadratic)

Factorising, (x - 7) (3x + 25) = 0,

either x = 7 or x = -25/3 ( but x cannot be negative)

Therefore, x = 7.