Define $a * b$ as $2a - b^2$. If $a * 5 = 9$, what is the value of $a$?
This problem involves a non-standard operation, so I need to follow the definition carefully. I know that $a * 5 = 9$, so I can plug in $5$ for $b$ in the definition and get $2a - 5^2 = 9$. Simplifying, I get $2a - 25 = 9$. Adding $25$ to both sides, I get $2a = 34$. Dividing both sides by $2$, I get $a = 17$. 