Problem 36 - Elementary and Intermediate Algebra

Question

Perform the operations. Simplify, if possible. $$ \frac{t+5}{t-5}-\frac{t-5}{t+5} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{20t}{(t-5)(t+5)}\).

Step 1: Identify the common denominator

To subtract these two fractions, we need a common denominator. The denominators are \(t-5\) and \(t+5\). The common denominator is the product of these denominators, \((t-5)(t+5)\).

Step 2: Rewrite each fraction with the common denominator

Multiply both the numerator and denominator of the first fraction by \(t+5\), and the second fraction by \(t-5\):\[\frac{t+5}{t-5} \cdot \frac{t+5}{t+5} = \frac{(t+5)^2}{(t-5)(t+5)}\]\[\frac{t-5}{t+5} \cdot \frac{t-5}{t-5} = \frac{(t-5)^2}{(t+5)(t-5)}\]

Step 3: Subtract the numerators

Both fractions now have the same denominator, so we can subtract the numerators directly:\[\frac{(t+5)^2 - (t-5)^2}{(t-5)(t+5)}\]

Step 4: Expand the squared terms

Expand \((t+5)^2\) and \((t-5)^2\) using the formula \((a+b)^2 = a^2 + 2ab + b^2\):\[(t+5)^2 = t^2 + 10t + 25\]\[(t-5)^2 = t^2 - 10t + 25\]

Step 5: Simplify the numerator

Subtract the expanded forms of \((t+5)^2\) and \((t-5)^2\):\[(t^2 + 10t + 25) - (t^2 - 10t + 25) = t^2 + 10t + 25 - t^2 + 10t - 25 = 20t\]

Step 6: Write the final simplified expression

Place the simplified numerator over the common denominator:\[\frac{20t}{(t-5)(t+5)}\]This is the simplified form of the original expression.

Key Concepts

Common Denominator

When dealing with algebraic fractions, finding a common denominator is one of the first steps before adding or subtracting them. The common denominator helps to bring two or more fractions to a common base so they can be easily combined. In our example, we have the fractions \(\frac{t+5}{t-5}\) and \(\frac{t-5}{t+5}\). Each has different denominators: \(t-5\) and \(t+5\). To find a common denominator, you multiply the denominators together. This gives us

• \((t-5)(t+5)\)

This means both fractions will share the same new denominator, allowing us to move forward with subtraction. Remember, finding the right common denominator simplifies the process significantly and prevents mistakes.

Simplifying Expressions

Simplifying expressions is a crucial part of solving algebraic fractions effectively. After identifying the common denominator for our fractions, the next step is rewriting both fractions with the new denominator. We multiplied the \(\frac{t+5}{t-5}\) fraction by \(\frac{t+5}{t+5}\) and the \(\frac{t-5}{t+5}\) fraction by \(\frac{t-5}{t-5}\).This gives us:

• \(\frac{(t+5)^2}{(t-5)(t+5)}\) for the first fraction
• \(\frac{(t-5)^2}{(t+5)(t-5)}\) for the second fraction

After this rewriting, both fractions now have the common denominator.To simplify further, expand the squared terms. The formula \((a+b)^2=a^2+2ab+b^2\) comes in handy here:

• \((t+5)^2 = t^2 + 10t + 25\)
• \((t-5)^2 = t^2 - 10t + 25\)

The simplification reveals the core expression behind those squared terms. It's a must to understand the expanded form to proceed effectively.

Subtracting Fractions

Once both fractions have a common denominator, the subtraction process becomes straightforward. With the known expressions:

• \((t+5)^2 = t^2 + 10t + 25\)
• \((t-5)^2 = t^2 - 10t + 25\)

We place them over the common denominator \((t-5)(t+5)\). The next step is to subtract the numerators: \[ (t^2 + 10t + 25) - (t^2 - 10t + 25) \]When performing subtraction, make sure to change the sign of every term in the second bracket before combining like terms. That results in:\[ 20t \]Now, place this simplified result over the common denominator:\[\frac{20t}{(t-5)(t+5)}\]This is the simplified version of the original expression. Make sure to verify by checking each step of subtraction to avoid errors in signs, making the difference clear and neat.