+1418 Let $x \mathbin{\spadesuit} y = \frac{x^2}{y}$ for all $x$ and $y$ such that $y\neq 0$. Find all values of $a$ such that $a \mathbin{\spadesuit} (a + 1) = 9$. Write your answer as a list separated by commas.
+129771 \($x \mathbin{\spadesuit} y = \frac{x^2}{y}$ for all $x$ and $y$ such that $y\neq 0$ \)
\(Find all values of $a$ such that $a \mathbin{\spadesuit} (a + 1) = 9$. \)
a (spade) (a + 1) = a^2 / (a + 1) = 9
a^2 = 9(a + 1)
a^2 = 9a + 9
a^2 - 9a - 9 = 0
a^2 - 9a = 9 complete the square on a
a^2 - 9a + 81/4 = 9 + 81/4
(a - 9/2)^2 = 117 / 4 take both roots
a - 9/2 = sqrt (117) / 2
a = [ 9 + sqrt (117) ] / 2 or a = [ 9 - sqrt (117) ] / 2
