Algebra

Let  $x\mathbin{\spadesuit}y = x^2/y$ for all $x$ and $y$ such that $y\neq 0$. Find all values of $a$ such that $a \mathbin{\spadesuit} (a + 1) = 9$. List the values you find in increasing order, separated by commas.

Alright, so in this case, we have $x = a$ and $y = a +1$. Putting into the form of $x\mathbin{\spadesuit}y = x^2/y$, we have $a \mathbin{\spadesuit} (a + 1) = \frac{a^2}{a+1} = 9$. 

Multiplying both sides by a + 1 and distributing the 9 in, we have $a^2 = 9a + 9$. Now we can write the quadratic $a^2 - 9a - 9 = 0$. 

Unforunately, we can't factor this directly, so we have to use the quadratic formula to find that

$a=\frac{3\sqrt{13}+9}{2}\\ a=\frac{-3\sqrt{13}+9}{2}$

This means our final answer is $a=\frac{3\sqrt{13}+9}{2}, \frac{-3\sqrt{13}+9}{2}$

Thanks!