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.
x♠y=x2y
$a♠(a+1)=9
So, using substitution
(a)^2 / (a + 1) = 9
a^2 = 9a + 9
a^2 -9a = 9 complete the square
a^2 - 9a + 81/4 = 9 + 81/4
(a - 9/2)^2 = 117 /4
a -9/2 = +/- sqrt [117] /2
a = [ 9 + sqrt (117) ] / 2 and a = [ 9 - sqrt (117) ] / 2