Sure! Let's go through the problem step-by-step and simplify the expressions to verify the given equation.
You are given the equation:
[tex]\[\frac{a^2 - 3ab + 2b^2}{a^2 - 4b^2} - \frac{4ab^2}{4a^2b + 8ab} = \frac{a - b}{a + b}\][/tex]
Let's simplify each part of the left-hand side of the equation.
### Step 1: Simplify each fraction
First fraction:
[tex]\[\frac{a^2 - 3ab + 2b^2}{a^2 - 4b^2}\][/tex]
Factorize the denominator and numerator if possible:
- The denominator [tex]\(a^2 - 4b^2\)[/tex] is a difference of squares, which can be factorized as:
[tex]\[a^2 - 4b^2 = (a - 2b)(a + 2b)\][/tex]
- The numerator [tex]\(a^2 - 3ab + 2b^2\)[/tex] does not factorize easily into simpler products.
Thus, we have:
[tex]\[\frac{a^2 - 3ab + 2b^2}{(a - 2b)(a + 2b)}\][/tex]
Second fraction:
[tex]\[\frac{4ab^2}{4a^2b + 8ab}\][/tex]
The denominator and numerator can be simplified by factoring out the common term:
- The numerator is already in its simplest form.
- The denominator can be rewritten as:
[tex]\[4a^2b + 8ab = 4ab(a + 2)\][/tex]
Thus, we have:
[tex]\[\frac{4ab^2}{4ab(a + 2)} = \frac{b}{a + 2}\][/tex]
Hence, the left-hand side expression can be rewritten as:
[tex]\[\frac{a^2 - 3ab + 2b^2}{(a - 2b)(a + 2b)} - \frac{b}{a + 2}\][/tex]
### Step 2: Common Denominator
To combine these two fractions, find a common denominator, which would be [tex]\((a - 2b)(a + 2b)\)[/tex]:
For the second fraction, [tex]\(\frac{b}{a + 2}\)[/tex], we need to multiply top and bottom by [tex]\((a - 2b)\)[/tex]:
[tex]\[\frac{b}{a + 2} \cdot \frac{a - 2b}{a - 2b} = \frac{b(a - 2b)}{(a - 2b)(a + 2b)}\][/tex]
Now we can combine the fractions:
[tex]\[\frac{a^2 - 3ab + 2b^2 - b(a - 2b)}{(a - 2b)(a + 2b)} = \frac{a^2 - 3ab + 2b^2 - ab + 2b^2}{(a - 2b)(a + 2b)} = \frac{a^2 - 4ab}{(a - 2b)(a + 2b)}\][/tex]
Now, note that:
[tex]\[a^2 - 4ab = a(a - 4b)\][/tex]
Hence, we have:
[tex]\[\frac{a(a - 4b)}{(a - 2b)(a + 2b)}\][/tex]
### Step 3: Check Right-Hand Side
We are given that the right-hand side of the equation is:
[tex]\[\frac{a - b}{a + b}\][/tex]
### Step 4: Verify the Equation
Given our simplifications and combining the fractions, we have the left-hand side as:
[tex]\[\frac{a(a - 4b)}{(a - 2b)(a + 2b)}\][/tex]
However, this expression is different from the right-hand side expression [tex]\(\frac{a - b}{a + b}\)[/tex]. Therefore, the original assertion:
[tex]\[\frac{a^2 - 3ab + 2b^2}{a^2 - 4b^2} - \frac{4ab^2}{4a^2b + 8ab} = \frac{a - b}{a + b}\][/tex]
is not correct.
The left-hand side simplifies to [tex]\(\frac{a(a - 4b)}{(a - 2b)(a + 2b)}\)[/tex], which does not equal the right-hand side.
### Conclusion
The detailed step-by-step solution shows that the simplified left-hand side expression does not match the right-hand side expression. Hence, the given equation is not valid.
