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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.

 Nov 26, 2023
 #1
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From the definition, [ a \mathbin{\spadesuit} (a + 1) = \frac{a^2}{a+1} = 9. ] Multiplying both sides by a+1 yields a2=9(a+1). Substituting a+1 for x and 9 for y into the definition of ♠, we find that this simplifies to 9(a+1)2​=9, which leads to a+1=±3. Therefore, a=−2,2​.

 Nov 27, 2023

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