SOLUTION: How to solve the following problems (with exact solutions): (2)/(x-1)+x= 5 and (2)/(x-1)+x= 2 (This has to do with rational functions)

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Question 825190: How to solve the following problems (with exact solutions):
(2)/(x-1)+x= 5 and (2)/(x-1)+x= 2
(This has to do with rational functions)
Answer by josgarithmetic(39623) About Me  (Show Source):
You can put this solution on YOUR website!
Rational EQUATIONS more so than functions.

First equation: Multiply left and right sides by x-1, and simplify; solve for x. Watch for a possible extraneous solution. You would have a quadratic equation.

Second equation: Similar to the first equation, multiply left and right sides by x-1, giving you a quadratic equation. Solve for x; watch for a possible extraneous solution.

The multiplication of both sides by the denominator clears the rational expressions.

Follow this generalization:
highlight_green%28n%2F%28x-a%29%2Bx=k%29
n%2Bx%28x-a%29=k%28x-a%29
n%2Bx%5E2-ax=kx-ka
x%5E2-ax-kx=-n-ka
x%5E2-%28a%2Bk%29x=-%28n-ka%29
highlight%28x%5E2-%28a%2Bk%29x%2Bn-ka=0%29
OR
highlight%28x%5E2-%28a%2Bk%29x%2B%28n-ka%29=0%29
Depending on the values you have for a, n, and k, you would either finish by factoring or finish by general solution to the quadratic formula.