To determine the two rational expressions whose difference completes the given equation, let's break down and analyze the expressions.
The given equation is:
[tex]\[ \frac{1}{x^2+6x} - \frac{x+2}{x^2-36} = \frac{x^2+x+6}{x(x-6)(x+6)} \][/tex]
### Consider Each Part Separately:
1. Expression on the Left:
[tex]\[ \frac{1}{x^2 + 6x} \][/tex]
We can simplify the denominator:
[tex]\[ x^2 + 6x = x(x + 6) \][/tex]
Therefore:
[tex]\[ \frac{1}{x(x + 6)} \][/tex]
2. Expression on the Right:
[tex]\[ \frac{x+2}{x^2-36} \][/tex]
We can factorize the denominator:
[tex]\[ x^2 - 36 = (x - 6)(x + 6) \][/tex]
Therefore:
[tex]\[ \frac{x+2}{(x-6)(x+6)} \][/tex]
### Perform the Subtraction:
The goal is to rewrite and combine these fractions so that they have a common denominator, which is:
[tex]\[ x(x-6)(x+6) \][/tex]
So we rewrite:
1. [tex]\[ \frac{1}{x(x + 6)} \rightarrow \frac{1 \cdot (x - 6)}{x(x + 6)(x - 6)} = \frac{x - 6}{x(x + 6)(x - 6)} = \frac{x - 6}{x(x - 6)(x + 6)} \][/tex]
2. [tex]\[ \frac{x+2}{(x-6)(x+6)} \rightarrow \frac{(x + 2) \cdot x}{x(x - 6)(x + 6)} = \frac{x(x + 2)}{x(x - 6)(x + 6)} = \frac{x(x + 2)}{x(x - 6)(x + 6)} \][/tex]
So now subtract:
[tex]\[ \frac{x - 6}{x(x - 6)(x + 6)} - \frac{x(x + 2)}{x(x - 6)(x + 6)} \][/tex]
Combine the numerators:
[tex]\[ \frac{(x - 6) - x(x + 2)}{x(x - 6)(x + 6)} \][/tex]
Expand and simplify the numerator:
[tex]\[ (x - 6) - x(x + 2) \][/tex]
[tex]\[ = x - 6 - x^2 - 2x \][/tex]
[tex]\[ = -x^2 - x - 6 \][/tex]
So we have:
[tex]\[ \frac{-x^2 - x - 6}{x(x - 6)(x + 6)} \][/tex]
Finally:
[tex]\[ \frac{x^2 + x + 6}{x(x - 6)(x + 6)} \][/tex]
Thus, the rational expressions that, when subtracted, complete the equation are:
[tex]\[ \boxed{\frac{1}{x^2 + 6x} \text{ and } \frac{x+2}{x^2 - 36}} \][/tex]
