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Algebra

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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.

May 4, 2024

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Let  $$x\mathbin{\spadesuit}y = x^2/y$$ for all $$x$$ and $$y$$ such that $$y\neq 0$$. Find all values of $$a$$ such that $$a \mathbin{\spadesuit} (a + 1) = 9$$. List the values you find in increasing order, separated by commas.

Alright, so in this case, we have $$x = a$$ and $$y = a +1$$. Putting into the form of $$x\mathbin{\spadesuit}y = x^2/y$$, we have $$a \mathbin{\spadesuit} (a + 1) = \frac{a^2}{a+1} = 9$$

Multiplying both sides by a + 1 and distributing the 9 in, we have $$a^2 = 9a + 9$$. Now we can write the quadratic $$a^2 - 9a - 9 = 0$$

Unforunately, we can't factor this directly, so we have to use the quadratic formula to find that

$$a=\frac{3\sqrt{13}+9}{2}\\ a=\frac{-3\sqrt{13}+9}{2}$$

This means our final answer is $$a=\frac{3\sqrt{13}+9}{2}, \frac{-3\sqrt{13}+9}{2}$$

Thanks!

May 4, 2024