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In place value notation, each column left, starting from the units column, is worth the base times the column on its right.
In base 10 this gives 1, 10, 10², ...
In base 8 this gives 1, 8, 8², ...
In base x this gives 1, x, x², ...
To convert a number in any base to base 10, multiply each digit by its place value column; this gives:
2x² + x + 1 = 1×8² + 5×8 + 3
→ 2x² + x + 1 = 64 + 40 + 3
→ 2x² + x + 1 = 107
→ 2x² + x - 106 = 0
→ x = (-1 ±√(1² - 4×2×-106))/(2×2)
→ x = (-1 ±√849)/4
→ x = (-1 - √849)/4 or x = (-1 + √849)/4
→ x ≈ -7.5344 or x ≈ 7.0344
A negative base is unlikely, thus the base x is (-1 + √849)/4 (or approx 7.0344)
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