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Find all values of $a$ such that $\frac{a-3}{\sqrt{a}} = -\sqrt{a}$.

 May 23, 2023
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First, we can square both sides of the equation to get rid of the square root. This gives us:

(a - 3)^2 = -a

We can then expand the left-hand side to get:

a^2 - 6a + 9 = -a

Combining like terms, we get:

a^2 - 5a + 9 = 0

We can then factor this equation to get:

(a - 1)(a - 9) = 0

This means that a = 1 or a = 9.

However, we need to check both solutions to make sure they are valid. If we substitute a = 1 into the original equation, we get:

(1 - 3)/sqrt(1) = -sqrt(1)

This is not a valid equation, since the square root of 1 is 1 and 1 - 3 is not equal to -1.

If we substitute a = 9 into the original equation, we get:

(9 - 3)/sqrt(9) = -sqrt(9)

This is a valid equation, since the square root of 9 is 3 and 9 - 3 is equal to 6.

Therefore, the only solution to the equation is a = 9.

 May 23, 2023

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