Problem about numbers in different bases

Question

For what numbers of $b$ is $100_b=10_{4b}$?

The answer says $b$=4

Can someone derive why? I dont get this I get:

$b^2=4b=40+b$

which is a second order equation?

Answer 1

$100_b = 10_{4b}$ means that $1\times b^2+0 \times b^1 + 0 \times 1= 1\times (4b)^1 + 0\times 1$, ie $b^2 = 4b$

So, if you divide this relation by b, you get $b = 4$

Answer 2

This is a nice example where the concatenation of symbols is potentially ambiguous. In particular, where the question writes "$4b$" for the base on the right hand side, it presumably has the interpretation $4$ times $b$. But the OP has interpreted it as two-digit decimal number whose unit digit is to be determined.