Question

perform the operation and combine to one fraction.
\\(\frac{x + 7}{x + 5} + \frac{3x}{x - 8}\\)
answer
attempt 1 out of 2

Explanation:

Step1: Find the common denominator

The denominators are \(x + 5\) and \(x - 8\), so the common denominator is \((x + 5)(x - 8)\).

Step2: Rewrite each fraction with the common denominator

For the first fraction \(\frac{x + 7}{x + 5}\), multiply numerator and denominator by \(x - 8\): \(\frac{(x + 7)(x - 8)}{(x + 5)(x - 8)}\)
For the second fraction \(\frac{3x}{x - 8}\), multiply numerator and denominator by \(x + 5\): \(\frac{3x(x + 5)}{(x + 5)(x - 8)}\)

Step3: Add the numerators

\(\frac{(x + 7)(x - 8)+3x(x + 5)}{(x + 5)(x - 8)}\)
Expand the numerators:
\((x + 7)(x - 8)=x^{2}-8x + 7x-56=x^{2}-x - 56\)
\(3x(x + 5)=3x^{2}+15x\)
Add them together: \(x^{2}-x - 56+3x^{2}+15x = 4x^{2}+14x - 56\)

Step4: Simplify the numerator (optional)

Factor out a 2 from the numerator: \(2(2x^{2}+7x - 28)\) (or leave as \(4x^{2}+14x - 56\))
So the combined fraction is \(\frac{4x^{2}+14x - 56}{(x + 5)(x - 8)}\) (or \(\frac{2(2x^{2}+7x - 28)}{(x + 5)(x - 8)}\))

Answer:

\(\frac{4x^{2}+14x - 56}{(x + 5)(x - 8)}\) (or simplified form \(\frac{2(2x^{2}+7x - 28)}{(x + 5)(x - 8)}\))