Certainly! To simplify the expression [tex]\(\frac{x}{x+3} + \frac{3x}{x-5}\)[/tex], we follow these steps:
1. Find a Common Denominator:
The denominators of the fractions are [tex]\(x + 3\)[/tex] and [tex]\(x - 5\)[/tex]. The common denominator will be the product of these two denominators:
[tex]\[
(x + 3)(x - 5)
\][/tex]
2. Rewrite Each Fraction with the Common Denominator:
We need to rewrite each fraction so that they both have the common denominator [tex]\((x + 3)(x - 5)\)[/tex].
- For the first fraction [tex]\(\frac{x}{x+3}\)[/tex], multiply the numerator and the denominator by [tex]\((x - 5)\)[/tex]:
[tex]\[
\frac{x \cdot (x - 5)}{(x + 3)(x - 5)} = \frac{x^2 - 5x}{(x + 3)(x - 5)}
\][/tex]
- For the second fraction [tex]\(\frac{3x}{x-5}\)[/tex], multiply the numerator and the denominator by [tex]\((x + 3)\)[/tex]:
[tex]\[
\frac{3x \cdot (x + 3)}{(x - 5)(x + 3)} = \frac{3x^2 + 9x}{(x - 5)(x + 3)}
\][/tex]
3. Add the Fractions:
Now that both fractions have the same denominator, we can add the numerators directly:
[tex]\[
\frac{x^2 - 5x + 3x^2 + 9x}{(x + 3)(x - 5)}
\][/tex]
4. Combine Like Terms in the Numerator:
Combine the [tex]\(x^2\)[/tex] terms and the [tex]\(x\)[/tex] terms in the numerator:
[tex]\[
x^2 - 5x + 3x^2 + 9x = 4x^2 + 4x
\][/tex]
5. Factor the Numerator if Possible:
Factor out the common factor from the numerator:
[tex]\[
4x^2 + 4x = 4x(x + 1)
\][/tex]
6. Write the Final Simplified Expression:
Place the simplified numerator over the common denominator:
[tex]\[
\frac{4x(x + 1)}{(x + 3)(x - 5)}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\frac{x}{x+3} + \frac{3x}{x-5}\)[/tex] is:
[tex]\[
\frac{4x(x + 1)}{(x + 3)(x - 5)}
\][/tex]
