Question

Find the difference of the following rational expressions find the missing part
x^2-2/x^2-5 - 4x+1/5-x^2+ ?/x^2-5

Answer 1

To find the missing part in the expression

\[
\frac{x^2 - 2}{x^2 - 5} - \frac{4x + 1}{5 - x^2} + \frac{?}{x^2 - 5},
\]

we first notice that \(5 - x^2\) can be rewritten as \(-(x^2 - 5)\). Thus, we can rewrite the second fraction:

\[
\frac{4x + 1}{5 - x^2} = \frac{4x + 1}{-(x^2 - 5)} = -\frac{4x + 1}{x^2 - 5}.
\]

Now our expression looks like this:

\[
\frac{x^2 - 2}{x^2 - 5} + \frac{-(4x + 1)}{x^2 - 5} + \frac{?}{x^2 - 5}.
\]

Combining these fractions gives:

\[
\frac{x^2 - 2 - (4x + 1) + ?}{x^2 - 5}.
\]

Simplifying the numerator:

\[
x^2 - 2 - 4x - 1 + ? = x^2 - 4x - 3 + ?.
\]

Now we have the final expression:

\[
\frac{x^2 - 4x - 3 + ?}{x^2 - 5}.
\]

To find the missing part such that this expression equals zero (the result of the difference of the two original fractions), we set:

\[
x^2 - 4x - 3 + ? = 0.
\]

This means we need to solve for ?:

\[
? = - (x^2 - 4x - 3).
\]

Thus, the missing part is:

\[
? = -x^2 + 4x + 3.
\]

Therefore, the final answer for the missing part is:

\[
\boxed{-x^2 + 4x + 3}.
\]