Problem 36 - College Algebra Essentials

Question

Add or subtract as indicated. $$\frac{x^{2}-4 x}{x^{2}-x-6}+\frac{4 x-4}{x^{2}-x-6}$$

Step-by-Step Solution

Answer

The final simplified form of the given expression is: \(x - 3\)

Step 1: Understand the Problem

Here, we have two fractions with the same denominators that need to be added together. In regular arithmetic, this could be done simply by adding the numbers in the numerator. The same principle applies in algebra as well. So, we proceed to add the numerators of the given fractions.

Step 2: Perform the Addition

Add the numerators together: \((x^{2}-4 x) + (4 x-4)\). This simplifies to \(x^{2} - 4x + 4x - 4\), and further simplification yields to \(x^{2} - 4\).

Step 3: Write the Final Answer

Now, we place the simplified numerator over the initially given denominator, resulting in \(\frac{x^{2} - 4}{x^{2}-x-6}\).

Step 4: Check for Further Simplifications

Check to see if the result can be simplified further. As \((x^{2}-4)\) contains a difference of squares that can be factored as \((x-2)(x+2)\) and \((x^{2}-x-6)\) can be factored as \((x-3)(x+2)\). Notice that they both have common factor \((x+2)\), hence our fractions simplifies to be: \(x - 3\).