To solve the given equation
[tex]\[
\frac{a^2 - 3ab + 2b^2}{a^2 - 4b^2} - \frac{4ab^2}{4a^2b + 8ab^2} = \frac{a - 2b}{a + 2b},
\][/tex]
let's go through a step-by-step process to simplify each fraction and compare both sides.
1. Simplify the first fraction:
[tex]\[
\frac{a^2 - 3ab + 2b^2}{a^2 - 4b^2}
\][/tex]
- The numerator [tex]\(a^2 - 3ab + 2b^2\)[/tex] can be factored as [tex]\((a - 2b)(a - b)\)[/tex].
- The denominator [tex]\(a^2 - 4b^2\)[/tex] is a difference of squares and can be factored as [tex]\((a - 2b)(a + 2b)\)[/tex].
Thus, the fraction simplifies to:
[tex]\[
\frac{(a - 2b)(a - b)}{(a - 2b)(a + 2b)}.
\][/tex]
If [tex]\(a \neq 2b\)[/tex],
[tex]\[
\frac{(a - b)}{(a + 2b)}.
\][/tex]
2. Simplify the second fraction:
[tex]\[
\frac{4ab^2}{4a^2b + 8ab^2}
\][/tex]
- The numerator [tex]\(4ab^2\)[/tex] remains as it is.
- The denominator [tex]\(4a^2b + 8ab^2\)[/tex] can be factored as [tex]\(4ab(a + 2b)\)[/tex].
Thus, the fraction simplifies to:
[tex]\[
\frac{4ab^2}{4ab(a + 2b)}.
\][/tex]
Simplifying this we get,
[tex]\[
\frac{b}{a + 2b}.
\][/tex]
3. Combine the simplified fractions:
Combining the simplified fractions from steps 1 and 2, we get:
[tex]\[
\frac{a - b}{a + 2b} - \frac{b}{a + 2b}.
\][/tex]
Since both fractions have the same denominator, we can combine them:
[tex]\[
\frac{a - b - b}{a + 2b} = \frac{a - 2b}{a + 2b}.
\][/tex]
4. Comparison with the right-hand side:
The right-hand side of the original equation is:
[tex]\[
\frac{a - 2b}{a + 2b}.
\][/tex]
Since we have simplified the left-hand side to [tex]\(\frac{a - 2b}{a + 2b}\)[/tex] and the right-hand side is also [tex]\(\frac{a - 2b}{a + 2b}\)[/tex], both sides are equal.
Therefore, the given equation
[tex]\[
\frac{a^2-3 ab+2 b^2}{a^2-4 b^2} - \frac{4 ab^2}{4 a^2 b+8 a b^2} = \frac{a-2 b}{a+2 b}
\][/tex]
is true.
