Certainly! Let's simplify the given expression step by step. The expression to simplify is:
[tex]\[
\frac{b - c}{2} + \frac{c + a}{2} + \frac{a + b}{2 - c^2}
\][/tex]
### Step 1: Combine the first two fractions
First, we'll look at the fractions [tex]\(\frac{b - c}{2}\)[/tex] and [tex]\(\frac{c + a}{2}\)[/tex]. Since they have the same denominator, we can combine them:
[tex]\[
\frac{b - c + c + a}{2} = \frac{b + a}{2}
\][/tex]
So, the expression now becomes:
[tex]\[
\frac{b + a}{2} + \frac{a + b}{2 - c^2}
\][/tex]
### Step 2: Simplify the combined fraction
Now, let us address the new expression:
[tex]\[
\frac{b + a}{2} + \frac{a + b}{2 - c^2}
\][/tex]
To combine these two fractions, we need a common denominator. The common denominator of 2 and [tex]\(2 - c^2\)[/tex] is their product: [tex]\(2(2 - c^2)\)[/tex].
### Step 3: Express both fractions with the common denominator
For [tex]\(\frac{b + a}{2}\)[/tex], multiply numerator and denominator by [tex]\(2 - c^2\)[/tex]:
[tex]\[
\frac{(b + a)(2 - c^2)}{2(2 - c^2)} = \frac{(b + a)(2 - c^2)}{2(2 - c^2)}
\][/tex]
For [tex]\(\frac{a + b}{2 - c^2}\)[/tex], multiply numerator and denominator by 2:
[tex]\[
\frac{2(a + b)}{2(2 - c^2)}
\][/tex]
### Step 4: Combine the fractions
Now, we can combine the two fractions because they have the same denominator:
[tex]\[
\frac{(b + a)(2 - c^2) + 2(a + b)}{2(2 - c^2)}
\][/tex]
### Step 5: Simplify the numerator
Expand and simplify the numerator:
[tex]\[
(b + a)(2 - c^2) + 2(a + b)
\][/tex]
[tex]\[
= b(2 - c^2) + a(2 - c^2) + 2a + 2b
\][/tex]
[tex]\[
= 2b - b c^2 + 2a - a c^2 + 2a + 2b
\][/tex]
Combine like terms:
[tex]\[
= 4b + 4a - b c^2 - a c^2
\][/tex]
We can factor out -1 from terms containing [tex]\(c^2\)[/tex]:
[tex]\[
= 4b + 4a - (b + a) c^2
\][/tex]
### Step 6: Write the final simplified expression
Therefore, the simplified expression is:
[tex]\[
\frac{4b + 4a - (b + a) c^2}{2(2 - c^2)}
\][/tex]
We can further simplify by factoring out the common terms in the numerator:
[tex]\[
\frac{4(a + b) - (a + b)c^2}{2(2 - c^2)}
\][/tex]
Now, factor out [tex]\( (a + b) \)[/tex] from the numerator:
[tex]\[
\frac{(a + b)(4 - c^2)}{2(2 - c^2)}
\][/tex]
Divide both numerator and denominator by 2:
[tex]\[
\frac{(a + b)(2 - \frac{c^2}{2})}{2 - c^2}
\][/tex]
Recognize that [tex]\[ 2 - \frac{c^2}{2} =(2-\frac{c^2}{2})\][/tex] can be written. Our simplified expression becomes:
[tex]\[
\frac{(-a - b + (a + b)(c^2 - 2)/2)}{c^2 - 2}
\][/tex]
This is indeed the simplified form of the given expression.
