Question

What is X when 211 base X is equal to 152 base 8?

Answer 1

$211_x=2x^2+x+1$
$152_8=1\cdot8^2+5\cdot8+2=106_{10}$

$2x^2+x+1=106$
$2x^2+x-105=0$
$2\left(x+\dfrac{15}2\right)(x-7)=0$

$\implies x=-\dfrac{15}2\text{ or }x=7$

Presumably $x$ is a non-negative integer, which leaves $x=7$, though it's not to say that rational solutions aren't necessarily excluded.