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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 27, 2024
 #1
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\( x \mathbin{\spadesuit} y = \frac{x^2}{y} \)

 

Find a such that   \(a \mathbin{\spadesuit} (a + 1) = 9\)

 

a^ 2  / ( a + 1)   = 9

 

a^2  = 9 (a + 1)

 

a^2 =  9a + 9

 

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  - 9/2  =  -sqrt (117) /  2

 

a =   (9 + 3sqrt (13)) / 2                a = (9 - 3sqrt (13))  / 2

 

 

cool cool cool

 Jan 28, 2024

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