Question

which expression is equivalent to \\(\frac{6}{x - 3} + \frac{2x}{x + 7}\\)? \\(\frac{-6x + 42}{x - 21}\\) \\(\frac{2x + 6}{2x - 4}\\) \\(\frac{2x^2 + 42}{x^2 + 4x - 21}\\) \\(\frac{2x^2 + 20x - 18}{x^2 + 4x - 21}\\) calculator

Explanation:

Step1: Find common denominator

The denominators are $x-3$ and $x+7$, so the common denominator is $(x-3)(x+7) = x^2 + 4x -21$.

Step2: Rewrite fractions with common denominator

$$\frac{6(x+7)}{(x-3)(x+7)} + \frac{2x(x-3)}{(x-3)(x+7)}$$

Step3: Expand numerators

$$\frac{6x + 42}{x^2+4x-21} + \frac{2x^2 - 6x}{x^2+4x-21}$$

Step4: Combine fractions and simplify numerator

$$\frac{6x + 42 + 2x^2 - 6x}{x^2+4x-21} = \frac{2x^2 + 42}{x^2+4x-21}$$

Answer:

$\boldsymbol{\frac{2x^2+42}{x^2+4x-21}}$ (third option)