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To solve the problem of adding the fractions [tex]\(\frac{2}{x-4} + \frac{x+7}{x+4}\)[/tex], we need to find a common denominator and then combine the numerators over that common denominator. Here’s the detailed step-by-step process:
1. Identify the Denominators:
The denominators are [tex]\(x - 4\)[/tex] and [tex]\(x + 4\)[/tex].
2. Find the Least Common Denominator (LCD):
Since the denominators [tex]\(x - 4\)[/tex] and [tex]\(x + 4\)[/tex] are relatively prime (they have no common factors other than 1), the least common denominator (LCD) is simply the product of these two factors:
[tex]\[ \text{LCD} = (x - 4)(x + 4) = x^2 - 16 \][/tex]
Here, we use the difference of squares to simplify the denominator: [tex]\((x - 4)(x + 4) = x^2 - 16\)[/tex].
3. Adjust the Numerators:
To add the fractions, we need to adjust each numerator, so they are expressed over the common denominator [tex]\(x^2 - 16\)[/tex].
- For [tex]\(\frac{2}{x-4}\)[/tex], multiply numerator and denominator by [tex]\(x + 4\)[/tex]:
[tex]\[ \frac{2}{x - 4} \times \frac{x + 4}{x + 4} = \frac{2(x + 4)}{(x - 4)(x + 4)} = \frac{2x + 8}{x^2 - 16} \][/tex]
- For [tex]\(\frac{x+7}{x+4}\)[/tex], multiply numerator and denominator by [tex]\(x - 4\)[/tex]:
[tex]\[ \frac{x + 7}{x + 4} \times \frac{x - 4}{x - 4} = \frac{(x + 7)(x - 4)}{(x + 4)(x - 4)} = \frac{x^2 + 7x - 4x - 28}{x^2 - 16} = \frac{x^2 + 3x - 28}{x^2 - 16} \][/tex]
4. Combine the Fractions:
Now that both fractions have a common denominator, we can add the numerators:
[tex]\[ \frac{2x + 8}{x^2 - 16} + \frac{x^2 + 3x - 28}{x^2 - 16} = \frac{(2x + 8) + (x^2 + 3x - 28)}{x^2 - 16} \][/tex]
Combine the terms in the numerator:
[tex]\[ (2x + 8) + (x^2 + 3x - 28) = x^2 + 5x - 20 \][/tex]
So, the combined fraction is:
[tex]\[ \frac{x^2 + 5x - 20}{x^2 - 16} \][/tex]
5. Simplify the Result:
We should check if the numerator and the denominator have any common factors and if the fraction can be simplified further. However, in this case, [tex]\(x^2 + 5x - 20\)[/tex] and [tex]\(x^2 - 16\)[/tex] have no common factors and cannot be simplified further. Thus, the fraction is already in its simplest form.
Therefore, the simplified rational expression for the given fractions [tex]\(\frac{2}{x-4} + \frac{x+7}{x+4}\)[/tex] is:
[tex]\[ \boxed{\frac{x^2 + 5x - 20}{x^2 - 16}} \][/tex]
1. Identify the Denominators:
The denominators are [tex]\(x - 4\)[/tex] and [tex]\(x + 4\)[/tex].
2. Find the Least Common Denominator (LCD):
Since the denominators [tex]\(x - 4\)[/tex] and [tex]\(x + 4\)[/tex] are relatively prime (they have no common factors other than 1), the least common denominator (LCD) is simply the product of these two factors:
[tex]\[ \text{LCD} = (x - 4)(x + 4) = x^2 - 16 \][/tex]
Here, we use the difference of squares to simplify the denominator: [tex]\((x - 4)(x + 4) = x^2 - 16\)[/tex].
3. Adjust the Numerators:
To add the fractions, we need to adjust each numerator, so they are expressed over the common denominator [tex]\(x^2 - 16\)[/tex].
- For [tex]\(\frac{2}{x-4}\)[/tex], multiply numerator and denominator by [tex]\(x + 4\)[/tex]:
[tex]\[ \frac{2}{x - 4} \times \frac{x + 4}{x + 4} = \frac{2(x + 4)}{(x - 4)(x + 4)} = \frac{2x + 8}{x^2 - 16} \][/tex]
- For [tex]\(\frac{x+7}{x+4}\)[/tex], multiply numerator and denominator by [tex]\(x - 4\)[/tex]:
[tex]\[ \frac{x + 7}{x + 4} \times \frac{x - 4}{x - 4} = \frac{(x + 7)(x - 4)}{(x + 4)(x - 4)} = \frac{x^2 + 7x - 4x - 28}{x^2 - 16} = \frac{x^2 + 3x - 28}{x^2 - 16} \][/tex]
4. Combine the Fractions:
Now that both fractions have a common denominator, we can add the numerators:
[tex]\[ \frac{2x + 8}{x^2 - 16} + \frac{x^2 + 3x - 28}{x^2 - 16} = \frac{(2x + 8) + (x^2 + 3x - 28)}{x^2 - 16} \][/tex]
Combine the terms in the numerator:
[tex]\[ (2x + 8) + (x^2 + 3x - 28) = x^2 + 5x - 20 \][/tex]
So, the combined fraction is:
[tex]\[ \frac{x^2 + 5x - 20}{x^2 - 16} \][/tex]
5. Simplify the Result:
We should check if the numerator and the denominator have any common factors and if the fraction can be simplified further. However, in this case, [tex]\(x^2 + 5x - 20\)[/tex] and [tex]\(x^2 - 16\)[/tex] have no common factors and cannot be simplified further. Thus, the fraction is already in its simplest form.
Therefore, the simplified rational expression for the given fractions [tex]\(\frac{2}{x-4} + \frac{x+7}{x+4}\)[/tex] is:
[tex]\[ \boxed{\frac{x^2 + 5x - 20}{x^2 - 16}} \][/tex]
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