To find the difference between the two given expressions:
[tex]\[
\frac{x}{x^2 - 2x - 15} - \frac{4}{x^2 + 2x - 35}
\][/tex]
we need to go through a series of steps. Let’s simplify the problem step-by-step.
1. Factorize the Denominators:
First, factorize the denominators of both the fractions.
The denominator [tex]\(x^2 - 2x - 15\)[/tex] factors as:
[tex]\[
x^2 - 2x - 15 = (x - 5)(x + 3)
\][/tex]
The denominator [tex]\(x^2 + 2x - 35\)[/tex] factors as:
[tex]\[
x^2 + 2x - 35 = (x + 7)(x - 5)
\][/tex]
2. Rewrite the Expressions:
Using these factorizations, we can rewrite each fraction:
[tex]\[
\frac{x}{x^2 - 2x - 15} = \frac{x}{(x - 5)(x + 3)}
\][/tex]
[tex]\[
\frac{4}{x^2 + 2x - 35} = \frac{4}{(x + 7)(x - 5)}
\][/tex]
3. Common Denominator:
To subtract these two fractions, we need a common denominator. The common denominator for [tex]\((x-5)(x+3)\)[/tex] and [tex]\((x-5)(x+7)\)[/tex] is [tex]\((x-5)(x+3)(x+7)\)[/tex].
4. Rewrite Each Fraction with the Common Denominator:
Rewrite each fraction with the common denominator [tex]\((x - 5)(x + 3)(x + 7)\)[/tex].
For [tex]\(\frac{x}{(x-5)(x+3)}\)[/tex]:
[tex]\[
\frac{x}{(x-5)(x+3)} = \frac{x(x+7)}{(x - 5)(x + 3)(x + 7)}
\][/tex]
For [tex]\(\frac{4}{(x+7)(x-5)}\)[/tex]:
[tex]\[
\frac{4}{(x+7)(x-5)} = \frac{4(x+3)}{(x - 5)(x + 3)(x + 7)}
\][/tex]
5. Subtract the Fractions:
Now, subtract the two fractions:
[tex]\[
\frac{x(x+7) - 4(x+3)}{(x-5)(x+3)(x+7)}
\][/tex]
Simplify the numerator:
[tex]\[
x(x + 7) - 4(x + 3)
\][/tex]
[tex]\[
x^2 + 7x - 4x - 12
\][/tex]
[tex]\[
x^2 + 3x - 12
\][/tex]
6. Write the Final Expression:
Putting it all together, the difference is:
[tex]\[
\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}
\][/tex]
This is the simplified form of the difference between the given expressions.
Therefore, the difference between:
[tex]\[
\frac{x}{x^2-2x-15} - \frac{4}{x^2+2x-35}
\][/tex]
is:
[tex]\[
\frac{x^2 + 3x - 12}{(x-5)(x+3)(x+7)}
\][/tex]
