Question

What is the least common denominator of the equation 5/x+5−1/x^2+2x−15=4/x^2+2x−15?

Answer 1

To find the least common denominator (LCD) of the equation

\[
\frac{5}{x+5} - \frac{1}{x^2 + 2x - 15} = \frac{4}{x^2 + 2x - 15},
\]

we first need to factor any expressions in the denominators.

1. The second term and the third term both have the denominator \(x^2 + 2x - 15\). We can factor this expression:

\[
x^2 + 2x - 15 = (x + 5)(x - 3).
\]

Thus, we can rewrite the equation as:

\[
\frac{5}{x + 5} - \frac{1}{(x + 5)(x - 3)} = \frac{4}{(x + 5)(x - 3)}.
\]

2. Now we observe the denominators:

- The first term has a denominator of \(x + 5\),
- The second term has a denominator of \((x + 5)(x - 3)\),
- The third term (which is equivalent to the second term) also has a denominator of \((x + 5)(x - 3)\).

3. The least common denominator (LCD) must include each factor in the denominators at its highest power. Here, we have:

- From \(\frac{5}{x + 5}\): the denominator \(x + 5\),
- From \(\frac{1}{(x + 5)(x - 3)}\) and \(\frac{4}{(x + 5)(x - 3)}\): the factors \(x + 5\) and \(x - 3\).

4. Therefore, the least common denominator is:

\[
(x + 5)(x - 3).
\]

In conclusion, the least common denominator of the given equation is

\[
\boxed{(x + 5)(x - 3)}.
\]