Problem 47 - College Algebra

Question

Add or subtract as indicated. $$ \frac{x+5}{x-5}+\frac{x-5}{x+5} $$

Step-by-Step Solution

Answer

The simplified expression of \(\frac{x+5}{x-5}+\frac{x-5}{x+5}\) is \(\frac{2x^{2}+50}{(x-5)(x+5)}\).

Step 1: Identify the Least Common Denominator (LCD)

The LCD is found by multiplying the two different denominators together. So, for this problem the LCD would be \((x-5)(x+5)\).

Step 2: Rewrite the fractions with the LCD

Rewrite each fraction so that the LCD is the denominator. Multiply each fraction by an equivalent of 1 that will transform the denominator into the LCD. Do this by using the other fraction's denominator. This yields:\[\frac{(x+5)(x+5)}{(x-5)(x+5)} + \frac{(x-5)(x-5)}{(x+5)(x-5)} \]

Step 3: Simplify the numerators

Now, simplify the numerators by multiplying the expressions out. \[\frac{x^{2}+10x+25}{(x-5)(x+5)} + \frac{x^{2}-10x+25}{(x+5)(x-5)} \]

Step 4: Combine the fractions and simplify

Since the denominators are the same, you can add the numerators together to combine the fractions. \[\frac{x^{2}+10x+25 + x^{2}-10x+25}{(x-5)(x+5)} \]This simplifies to \[\frac{2x^{2}+50}{(x-5)(x+5)}\]

Key Concepts

Least Common Denominator - The Foundation of Adding Fractions

When facing the task of adding fractions, the least common denominator (LCD) is crucial for combining them into a single fraction. The LCD is the smallest number that is a multiple of both denominators in the given fractions. Think of it as the common ground where both fractions can come together. In practice, finding the LCD involves identifying the unique factors each denominator has and multiplying them to form the least shared multiple.

For instance, in our exercise, the denominators are \(x-5\) and \(x+5\). These two expressions do not share any common factors, so their LCD is simply their product: \(x-5)(x+5)\). By converting both fractions to have this shared denominator, we can then proceed to add or subtract them. It's like matching the gears of two different machines so they fit together perfectly and run as one. Without identifying the LCD first, you cannot accurately perform arithmetic on fractions.

Simplifying Numerators - Crafting Clearer Fractions

Once the fractions share a common denominator, our attention shifts to the simplifying numerators. Simplifying, in essence, means rewriting the numerators in their most streamlined form. This process often involves multiplying out any polynomials and combining like terms to make the fraction easier to work with and understand.

In the problem at hand, we simplify the numerators by expanding the products \(x+5)(x+5)\) and \(x-5)(x-5)\), which yields \(x^2+10x+25\) and \(x^2-10x+25\), respectively. Simplification makes the subsequent steps clearer and prevents potential errors. It’s akin to tidying up a room before you redecorate; it clears the clutter and provides a clean canvas to work on. This step is vital for ensuring the fractions are primed for addition or subtraction.

Combining Fractions - The Act of Unifying Parts

Once the fractions have a common denominator and the numerators are simplified, we are ready for combining fractions. This is where the preparation pays off: with the LCD established and numerators simplified, you can add up the numerators while keeping the denominator constant.

For the provided exercise, we combine the numerators \(x^2+10x+25\) and \(x^2-10x+25\) to obtain \(2x^2+50\). Combining fractions is much like gathering apples into a basket; you count them all up, but the basket—the denominator—remains the same. The skill of combining fractions underpins much of algebra and beyond. It’s a foundational tool in a mathematician's toolkit, simplifying complex expressions and paving the way for more advanced math concepts.