<s>[INST] What base system satisfies the equation For the equation:
$$5x^2 - 50x +125 = 0$$
$x=5$ and $x=8$ are solutions.
This is in another base, what are the steps required to find out what base it is in?
Thanks
 [/INST] Let $b$ be the base of the coefficients in the quadratic. Then, $(50)_b = 5b$ and $(125)_b = b^2+2b+5$. 
Hence, the quadratic in base-$10$ is $5x^2-5bx+(b^2+2b+5) = 0$. 
Since $x = 5$ is a solution, we have $125 - 25b + (b^2+2b+5) = 0$, i.e. $b^2-23b+130 = 0$.
Since $x = 8$ is a solution, we have $320 - 40b + (b^2+2b+5) = 0$, i.e. $b^2-38b+325 = 0$.
Can you find what value of $b$ satisfies both of these conditions?
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