Find all values of x such that 6 = 35/x - 49/x^2 + 5/x^2. If you find more than one value, then list your solutions in increasing order, separated by commas.
This question is mostly algebraic manipulation.
First, the equation can be simplified to 6 = 35/x - 44/x^2.
Multiplying by x^2 on both sides yields 6x^2 = 35x - 44.
Rearranging the equation leads to 6x^2 - 35x + 44 = 0.
Factoring the quadratic leads to (6x - 11)(x - 4) = 0.
Now, solve for both factors using the Zero Product Property.
6x - 11 = 0, so x = 11/6.
x - 4 = 0, so x = 4.
Since the original equation introduced fractions with variables in the denominator, I quickly verified that these solutions are correct, and they are.
11/6 < 4, so listing the solutions in increasing order yields 11/6, 4