Final answer:
The simplest form of the expression has 3x-4 in the numerator and x(x-2) in the denominator.
Step-by-step explanation:
The question involves simplifying a given algebraic fraction. Firstly, identify the common denominator, which in this case is x² - 4x + 4, a perfect square trinomial that can be factored to (x - 2)². After finding the common denominator, add or subtract the numerators accordingly. This will simplify the complex fraction to its simplest form.
To work through the expression 3/x-2 + x-2/x²-4x+4, factor the quadratic expression to get (x - 2)² in the denominator. Then, rewrite each term with this common denominator and combine the numerators. After simplifying, you may find that certain terms cancel out, leading to the simplest form of the expression with a specific numerator and denominator.
When performing operations on both sides of an equation, always maintain the equality by doing the same to both sides. During simplification, if any quantity is the same in the numerator and the denominator, it can be cancelled out since any number divided by itself equals one.
Using negative exponents appropriately can also simplify expressions, as they denote division instead of multiplication. This can change the location of variables and numbers within the fractions, potentially simplifying the expression further. Additionally, remember that in algebraic equations with one variable, we seek a single value that satisfies the equation; extraneous or non-sensible solutions are discarded.
If you come across quadratic equations within the simplification process, the quadratic formula may be necessary to find the values of x. However, remember that the context of the problem, such as ionization of weak acids, may allow for further simplification based on assumptions like the small extent of ionization.
The simplest form of the expression has 3x-4 in the numerator and x(x-2) in the denominator.