Question

If $a,$ $b,$ and $c$ are positive integers, the radicals $\sqrt{a+\dfrac bc}$ and $a\sqrt{\dfrac bc}$ are equal when and only when:

$a=b=c=1$

$a=b$ and $c=a=1$

$c=\dfrac{b(a^2-1)}a$

$a=b$ and $c$ is any value

$a=b$ and $c=a-1$

Both sides are positive, so square the equality

Multiply the resulting equation by $c$ and isolate $c$

Because all quantities are positive, squaring is reversible: $a+\frac bc=a^2\frac bc.$ Multiplying by $c$ gives $ac+b=a^2b,$ so $c=\frac{b(a^2-1)}a.$

Thus, the correct answer is C.