To solve the rational equation
[tex]\[
\frac{4}{a-2} - \frac{8}{a^2 - 2a} = -2,
\][/tex]
we first notice that [tex]\(a^2 - 2a\)[/tex] can be factored as [tex]\(a(a-2)\)[/tex]. Let's rewrite the equation with this factorization:
[tex]\[
\frac{4}{a-2} - \frac{8}{a(a-2)} = -2.
\][/tex]
Next, we need to find a common denominator for the fractions. The common denominator for [tex]\(a-2\)[/tex] and [tex]\(a(a-2)\)[/tex] is [tex]\(a(a-2)\)[/tex]. Rewriting each term with this common denominator, we get:
[tex]\[
\frac{4a}{a(a-2)} - \frac{8}{a(a-2)} = -2.
\][/tex]
Combine the fractions on the left-hand side:
[tex]\[
\frac{4a - 8}{a(a-2)} = -2.
\][/tex]
To eliminate the fraction, we multiply both sides of the equation by the common denominator [tex]\(a(a-2)\)[/tex]:
[tex]\[
4a - 8 = -2a(a-2).
\][/tex]
Distribute and simplify the right-hand side:
[tex]\[
4a - 8 = -2a^2 + 4a.
\][/tex]
Subtract [tex]\(4a\)[/tex] from both sides to isolate the quadratic term:
[tex]\[
-8 = -2a^2.
\][/tex]
Divide both sides by [tex]\(-2\)[/tex]:
[tex]\[
4 = a^2.
\][/tex]
Taking the square root of both sides, we find the solutions:
[tex]\[
a = \pm 2.
\][/tex]
We need to check these potential solutions in the original equation, because the value [tex]\(a = 2\)[/tex] makes the denominators zero, which is not allowed. Substituting [tex]\(a = 2\)[/tex] into the original equation, the denominators become zero, indicating a restriction. Thus, [tex]\(a = 2\)[/tex] is not a valid solution.
This leaves us with:
[tex]\[
a = -2.
\][/tex]
Thus, the solution set for the given equation is:
[tex]\[
\{-2\}.
\][/tex]
