Question

adding / subtracting rational expressions (lev
question
perform the operation and combine to one fraction.
\\(\frac{5}{x - 2} + \frac{3x + 1}{x + 3}\\)
answer
attempt 1 out of 2

Explanation:

Step1: Find the common denominator

The denominators are \(x - 2\) and \(x + 3\), so the common denominator is \((x - 2)(x + 3)\).

Step2: Rewrite each fraction with the common denominator

For \(\frac{5}{x - 2}\), multiply numerator and denominator by \(x + 3\): \(\frac{5(x + 3)}{(x - 2)(x + 3)}=\frac{5x + 15}{(x - 2)(x + 3)}\)
For \(\frac{3x + 1}{x + 3}\), multiply numerator and denominator by \(x - 2\): \(\frac{(3x + 1)(x - 2)}{(x - 2)(x + 3)}=\frac{3x^2 - 6x + x - 2}{(x - 2)(x + 3)}=\frac{3x^2 - 5x - 2}{(x - 2)(x + 3)}\)

Step3: Add the two fractions

\(\frac{5x + 15}{(x - 2)(x + 3)}+\frac{3x^2 - 5x - 2}{(x - 2)(x + 3)}=\frac{5x + 15 + 3x^2 - 5x - 2}{(x - 2)(x + 3)}\)
Simplify the numerator: \(5x - 5x + 15 - 2 + 3x^2 = 3x^2 + 13\)
So the combined fraction is \(\frac{3x^2 + 13}{(x - 2)(x + 3)}\) (we can also expand the denominator: \(\frac{3x^2 + 13}{x^2 + x - 6}\))

Answer:

\(\frac{3x^{2}+13}{x^{2}+x - 6}\) (or \(\frac{3x^{2}+13}{(x - 2)(x + 3)}\))