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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.

 Jan 29, 2025
 #1
avatar+130383 
+1

xy=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

 

cool cool cool

 Jan 29, 2025

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