Let's solve the expression step-by-step:
Given expression:
[tex]$\frac{m+b}{m-b} + \frac{m-b}{m+b} - \frac{m^2+b^2}{m^2-b^2}$[/tex]
1. Simplify each fraction:
- First, let’s rewrite the third term:
[tex]\[
\frac{m^2+b^2}{m^2-b^2}
\][/tex]
Notice that the denominator [tex]\( m^2 - b^2 \)[/tex] can be factored as [tex]\( (m - b)(m + b) \)[/tex].
- Therefore, the expression becomes:
[tex]\[
\frac{m^2 + b^2}{(m - b)(m + b)}
\][/tex]
2. Substitute the factored form back into the main expression:
[tex]\[
\frac{m+b}{m-b} + \frac{m-b}{m+b} - \frac{m^2+b^2}{(m - b)(m + b)}
\][/tex]
3. Combine all the terms over a common denominator:
- The common denominator for the first two terms is [tex]\( (m-b)(m+b) \)[/tex].
- Combining them all over the common denominator gives us:
[tex]\[
\frac{(m+b)(m+b) + (m-b)(m-b) - (m^2+b^2)}{(m-b)(m+b)}
\][/tex]
4. Expand the numerators of the first two fractions:
- Expanding [tex]\( (m+b)(m+b) \)[/tex]:
[tex]\[
(m+b)^2 = m^2 + 2mb + b^2
\][/tex]
- Expanding [tex]\( (m-b)(m-b) \)[/tex]:
[tex]\[
(m-b)^2 = m^2 - 2mb + b^2
\][/tex]
- Now the expression becomes:
[tex]\[
\frac{(m^2 + 2mb + b^2) + (m^2 - 2mb + b^2) - (m^2 + b^2)}{(m-b)(m+b)}
\][/tex]
5. Combine like terms in the numerator:
- Summing the terms in the numerator:
[tex]\[
(m^2 + 2mb + b^2) + (m^2 - 2mb + b^2) - (m^2 + b^2) \\
= m^2 + 2mb + b^2 + m^2 - 2mb + b^2 - m^2 - b^2
\][/tex]
- Simplifying:
[tex]\[
= (m^2 + m^2 - m^2) + (2mb - 2mb) + (b^2 + b^2 - b^2) \\
= m^2 - m^2 - b^2
= -(m^2 + b^2)
\][/tex]
6. Thus, the simplified form of the numerator is:
[tex]\[
-(m^2 + b^2)
\][/tex]
- So, the overall expression is now:
[tex]\[
\frac{-(m^2 + b^2)}{(m-b)(m+b)}
\][/tex]
7. This can be further simplified to:
[tex]\[
-\frac{m^2 + b^2}{m^2 - b^2}
\][/tex]
So, the fully simplified expression is:
[tex]\[
-\frac{m^2 + b^2}{m^2 - b^2}
\][/tex]
And in terms of the given result:
[tex]\[
(-b^2 - m^2)/(b^2 - m^2)
\][/tex]
Thus, our final simplified answer is:
[tex]\[
-\frac{b^2 + m^2}{b^2 - m^2}
\][/tex]
Hope this helps!
