Find all values of $a$ that satisfy the equation
\[\frac{a}{3} + 1 = \frac{a + 3}{a} - \frac{a^2 + 2}{a}.\]
what is so hard about just using regular math notation?
What we were given was
\[\frac{a}{3} + 1 = \frac{a + 3}{a} - \frac{a^2 + 2}{a}.\]
a/3 + 1 = (a+3)/a - (a^2+2)/a
multiplying through by 3a, that gives
a^2 + 3a = 3(a+3) - 3(a^2+2)
a^2 + 3a = 3a+9-3a^2-6
4a^2 = 3
a = ±√3/2
You are correct, and I apologize for the confusion. Here is the correct solution:
Starting with the equation
\[\frac{a}{3} + 1 = \frac{a + 3}{a} - \frac{a^2 + 2}{a},\]
we can multiply both sides by $3a$ to obtain
\[a^2 + 3a = 3(a + 3) - 3(a^2 + 2).\]
Simplifying the right side of the equation gives
\[a^2 + 3a = 3a + 9 - 3a^2 - 6.\]
Collecting like terms and rearranging, we have
\[4a^2 = 3.\]
Dividing both sides by $4$ gives
\[a^2 = \frac{3}{4}.\]
Taking the square root of both sides yields
\[a = \pm \frac{\sqrt{3}}{2}.\]
Therefore, the values of $a$ that satisfy the equation are $a = \boxed{\frac{\sqrt{3}}{2}, -\frac{\sqrt{3}}{2}}$.
To find all values of $a$ that satisfy the given equation, we first need to simplify it. We can start by removing the fractions by finding a common denominator.
The denominator for the second fraction is $a$, so we can multiply the numerator and denominator of the first fraction by $a$ to obtain:
\[\frac{a^2}{3} + a = \frac{(a + 3)(a)}{a} - \frac{a^2 + 2}{a}.\]
Simplifying further, we have:
\[\frac{a^2}{3} + a = a + 3 - \frac{a^2 + 2}{a}.\]
Next, we can clear the fractions by multiplying through by $3a$:
\[3a^2 + 3a = 3a^2 + 9a - (a^2 + 2).\]
Simplifying this equation, we get:
\[3a^2 + 3a = 3a^2 + 9a - a^2 - 2.\]
Combining like terms, we can rewrite the equation as:
\[0 = 7a - 2.\]
To solve for $a$, we isolate $a$ by moving the constant term to the right-hand side:
\[7a = 2.\]
Finally, we divide both sides of the equation by 7 to solve for $a$:
\[a = \frac{2}{7}.\]
Therefore, the only value of $a$ that satisfies the given equation is $a = \frac{2}{7}$.
Multiplying every term by $a$, we can rewrite the equation in a form more suitable for finding the values of $a$:
\[a^2 + a = a^2 + 2 - 3.\]
Canceling the $a^2$ terms leaves $a = -1$, so the only solution is $a = \boxed{-1}$.