Question

drag each expression to the correct location in the equation. not all expressions will be used.
determine the two rational expressions whose difference completes this equation.
\\(\frac{x - 2}{x^2 + 36}\\) \\(\frac{1}{x^2 + 6x}\\) \\(\frac{x + 2}{x^2 - 36}\\) \\(\frac{1}{x^2 + 6}\\)
\\(\square - \square = \frac{x^2 + x + 6}{x(x - 6)(x + 6)}\\)

Explanation:

Step1: Factor all denominators

• $\frac{-x-2}{x^2+36}$: denominator is prime
• $\frac{1}{x^2+6x}=\frac{1}{x(x+6)}$
• $\frac{x+2}{x^2-36}=\frac{x+2}{(x-6)(x+6)}$
• $\frac{1}{x^2+6}$: denominator is prime
• Right-hand side denominator: $x(x-6)(x+6)$

Step2: Test valid expression pairs

We test $\frac{x+2}{(x-6)(x+6)}$ and $\frac{1}{x(x+6)}$:
First, find a common denominator $x(x-6)(x+6)$:
$$\frac{x+2}{(x-6)(x+6)} = \frac{x(x+2)}{x(x-6)(x+6)} = \frac{x^2+2x}{x(x-6)(x+6)}$$
$$\frac{1}{x(x+6)} = \frac{(x-6)}{x(x-6)(x+6)} = \frac{x-6}{x(x-6)(x+6)}$$

Step3: Compute the difference

$$\frac{x^2+2x}{x(x-6)(x+6)} - \frac{x-6}{x(x-6)(x+6)} = \frac{(x^2+2x)-(x-6)}{x(x-6)(x+6)}$$
Simplify the numerator:
$$(x^2+2x)-(x-6) = x^2+2x-x+6 = x^2+x+6$$

Answer:

First box: $\boldsymbol{\frac{x+2}{x^2-36}}$, Second box: $\boldsymbol{\frac{1}{x^2+6x}}$