Hi Jake. I note your extra spacing around the second minus sign; yet, a lack of such spacing around operators is not sufficient to denote a denominator. As Denis implied, you must use grouping symbols for that.
In accordance with the Order of Operations, your typing represents the following non-quadratic.
\(\displaystyle \dfrac{x}{x} - 2 - \dfrac{2}{5} = \dfrac{1}{x} + 4\)
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x/(x - 2) - 2/5 = 1/(x + 4)
To rewrite this equation as a quadratic, you need to eliminate the ratios. This can be done in more than one way.
I will list the steps for one such way; you do the work, and show us what you get.
1) Add 2/5 to each side
2) Subtract 1/(x + 4) from each side
3) Multiply each side by 5
At this point, you ought to have a difference of ratios on the left-hand side and an integer on the right-hand side.
4) Apply the property known as 'cross multiplication' to the difference of ratios on the left-hand side.
If you're unsure of this property, here it is (shown symbolically):
A/B - C/D cross-multiplied yields
(A)(D) - (B)(C)
5) Multiply out everything on the left-hand side, and combine like-terms
6) Subtract the integer on the right-hand side from both sides
At this point, you ought to have a quadratic equation in the form of ax^2 + bx + c = 0
By the way, are you familiar with Complex numbers? Each of the two solutions for the given equation contain an imaginary part.
If you get stuck, please show how far you got, so that we may check your work in progress. Otherwise, let us know if you have any specific questions, before you proceed.
Cheers