91Ó°ÊÓ

When working with rational expressions, finding the least common denominator (LCD) is crucial for adding or subtracting these expressions. The LCD is essentially the smallest expression that both denominators can divide into without leaving a remainder. This helps in transforming the fractions to have a common base, making it easier to combine them.

For the expression \(\frac{2x}{x+2}+\frac{x+2}{x-2}\), the denominators are \(x+2\) and \(x-2\). To find the LCD, we multiply these two terms together, resulting in \((x+2)(x-2)\), which is equivalent to \(x^2 - 4\).

• Multiplying denominators helps cover both expressions completely.
• The LCD ensures no fractions are left with division remainders.

Finding the LCD enables us to rewrite each rational expression with this new denominator, setting the stage for further simplification.

After determining the least common denominator, the next step is simplifying the fractions. This means rewriting each fraction so they have the same denominator, making them compatible for addition.

Initially, we have:

• \(\frac{2x}{x+2}\)
• \(\frac{x+2}{x-2}\)

To rewrite these fractions using the LCD \((x+2)(x-2)\), we need to adjust the numerators accordingly:
1. Multiply \(2x\) by \(x-2\), resulting in \(2x(x-2) = 2x^2 - 4x\).
2. Multiply \(x+2\) by itself, resulting in \((x+2)(x+2) = x^2 + 4x + 4\).

Now, both fractions are expressed with the same denominator, \((x^2 - 4)\):

• \(\frac{2x^2 - 4x}{x^2 - 4}\)
• \(\frac{x^2 + 4x + 4}{x^2 - 4}\)

These simplifications make it straightforward to proceed with addition.

With both fractions simplified to have a common denominator, the final step is to add the numerators together. This process is similar to adding numbers over a common denominator in simple arithmetic.

Using the simplified forms:

• \(\frac{2x^2 - 4x}{x^2 - 4}\)
• \(\frac{x^2 + 4x + 4}{x^2 - 4}\)

Place them together under a single fraction line having the common denominator, and add the numerators: \[\frac{2x^2 - 4x + x^2 + 4x + 4}{x^2 - 4}\]
This gives \(\frac{3x^2 + 4}{x^2 - 4}\) as the simplified resultant fraction.

• Adding fractions involves simply combining the numerators.
• The denominator remains the same throughout the process.

Understanding these steps ensures you can handle similar problems with ease, without losing track of each part of the calculation.