Adopted from the Vermont Math contest

Algebra Level 3

Find the smallest positive real $x$ such that $\big\lfloor x^2 \big\rfloor-x\lfloor x \rfloor=6.$ If your answer is in the form $\frac{a}{b}$ , where $a$ and $b$ are coprime positive integers, submit your answer as $a+b.$

Notation: $\lfloor \cdot \rfloor$ denotes the floor function.