<s>[INST] Find the base numeric system Find the numeric base we are using if $ x = 4 $ and $ x = 7 $ are zeros of $ 5x^2 - 50x + 118. $
When I plug in $ x = 4 $ and $ x = 7, $ I receive $ -2 $ and $ 13, $ respectively, but how do I proceed from that?
 [/INST] Let $b$ be the base in question. Presumably $b>8$, since $8$ appears as a digit in $118$. We also know that $5x^2-50x+118$ is a multiple of $(x-4)(x-7)$, and since the $x^2$ term of $(x-4)(x-7)$ is simply $x^2$, we must have
$$5x^2-50x+118=5(x-4)(x-7)\;.$$
From this we deduce that $55$ in base ten is $50_b$. This clearly implies that $b=11$ (in base ten), and this is confirmed by the fact that $5\cdot4\cdot7=140$ in base ten is $118_b$: with $b=11$ this is $121+11+8=140$.
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