If (146)x + (313)x-2 = (246)8 , then the value of base x is-

First, convert the right-hand side (246)8 to decimal: 2(82) + 4(8) + 6 = 166. Next, express the left-hand side in terms of x. (146)x = x2 + 4x + 6 and (313)x-2 = 3(x-2)2 + (x-2) + 3 = 3x2 - 11x + 13. Summing these gives 4x2 - 7x + 19 = 166, which simplifies to 4x2 - 7x - 147 = 0. Solving this quadratic equation yields x=7 (discarding the negative root). Since digits in base x must be less than x, and the digit 6 appears, we require x > 6. Thus, x=7 is the valid solution.