Question

enter the values for the highlighted variables that show how to subtract the rational expressions correctly:
\\(\frac{2}{x^2 - 36} - \frac{1}{x^2 + 6x} = \frac{2}{(x + 6)(x - 6)} - \frac{1}{x(x + a)}\\)
\\(= \frac{bx}{(x + 6)(x - 6)x} - \frac{x - c}{(x + 6)(x - 6)x}\\)
\\(= \frac{dx - x + e}{(x + 6)(x - 6)x}\\)
\\(= \frac{x + f}{(x + 6)(x - 6)x}\\)
\\(= \frac{g}{x(x - 6)}\\)
\\(a = \square\\)
\\(b = \square\\)
\\(c = \square\\)
\\(d = \square\\)
\\(e = \square\\)
\\(f = \square\\)
\\(g = \square\\)

Explanation:

Step1: Factor denominator for $a$

Factor $x^2+6x = x(x+6)$, so $a=6$.

Step2: Find common denominator for $b$

Multiply numerator/denominator of $\frac{2}{(x+6)(x-6)}$ by $x$: $\frac{2x}{(x+6)(x-6)x}$, so $b=2$.

Step3: Adjust second fraction for $c$

Multiply numerator/denominator of $\frac{1}{x(x+6)}$ by $(x-6)$: $\frac{x-6}{(x+6)(x-6)x}$, so $c=6$.

Step4: Combine terms for $d$

From $\frac{2x - (x-6)}{(x+6)(x-6)x}$, $2x$ gives $d=2$.

Step5: Simplify numerator for $e$

$- (x-6) = -x + 6$, so $e=6$.

Step6: Simplify numerator for $f$

$2x - x + 6 = x + 6$, so $f=6$.

Step7: Cancel terms for $g$

Cancel $(x+6)$ from $\frac{x+6}{(x+6)(x-6)x}$: $\frac{1}{x(x-6)}$, so $g=1$.

Answer:

$a=6$
$b=2$
$c=6$
$d=2$
$e=6$
$f=6$
$g=1$