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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 9, 2024
 #2
avatar+128732 
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\($x \mathbin{\spadesuit} y = \frac{x^2}{y}$ for all $x$ and $y$ such that $y\neq 0$ \)

\(Find all values of $a$ such that $a \mathbin{\spadesuit} (a + 1) = 9$. \)

 

 

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

 

 

cool cool cool

 Jan 9, 2024

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