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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.

Jun 11, 2024

#1
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We want to solve the equation for

$$a\mathbin{\spadesuit} (a + 1) = 9$$

Now, we plug in the function, and we get the equation $$9=\frac{a^2}{a+1}$$

Now, we simplfy solve the equation. We get

$$9a+9=a^2$$

$$a^2-9a-9=0$$

Using the quadratic equation, we get

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

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

Thanks! :)

Jun 11, 2024
#2
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If $$x \mathbin{\spadesuit} y = \frac{x^2}{y}$$, then $$a \mathbin{\spadesuit} (a + 1) = \frac{a^2}{a+1}$$. Therfore, $$\frac{a^2}{a+1}=9$$. We can now use completing the square to solve for a.

$$\frac{a^2}{a+1}=9$$

$$a^2 = 9a+9$$

$$a^2-9a-9=0$$

$$a^2-9a+20.25 = 29.25$$

$$(a-4.5)^2 = 29 \frac{1}{4}$$

$$(a-\frac{9}{2})^2 = \frac{117}{4}$$

$$a-\frac{9}{2}=\frac{\pm\sqrt{117}}{2}$$

$$\mathbf{a = \frac{9\pm3\sqrt{13}}{2}}$$

So the two solutions are (9 + 3sqrt(13))/2, (9 - 3sqrt(13))/2

Jun 11, 2024
edited by Maxematics  Jun 11, 2024