To determine the difference between the two fractions [tex]\( \frac{x}{x^2 - 2x - 15} \)[/tex] and [tex]\( \frac{4}{x^2 + 2x - 35} \)[/tex], follow these steps:
1. Factorize the Denominators:
- For [tex]\( x^2 - 2x - 15 \)[/tex]:
[tex]\[
x^2 - 2x - 15 = (x - 5)(x + 3)
\][/tex]
- For [tex]\( x^2 + 2x - 35 \)[/tex]:
[tex]\[
x^2 + 2x - 35 = (x + 7)(x - 5)
\][/tex]
2. Determine the Common Denominator:
- The common denominator for both fractions will be the product of the two individual denominators:
[tex]\[
(x - 5)(x + 3)(x + 7)
\][/tex]
3. Adjust the Numerators:
- Adjust the numerator of the first fraction to have the common denominator:
[tex]\[
\frac{x}{(x - 5)(x + 3)} = \frac{x \cdot (x + 7)}{(x - 5)(x + 3)(x + 7)} = \frac{x(x + 7)}{(x - 5)(x + 3)(x + 7)}
\][/tex]
Simplify the numerator:
[tex]\[
x(x + 7) = x^2 + 7x
\][/tex]
So, it becomes:
[tex]\[
\frac{x^2 + 7x}{(x - 5)(x + 3)(x + 7)}
\][/tex]
- For the second fraction:
[tex]\[
\frac{4}{(x + 7)(x - 5)} = \frac{4 \cdot (x + 3)}{(x - 5)(x + 3)(x + 7)} = \frac{4(x + 3)}{(x - 5)(x + 3)(x + 7)}
\][/tex]
Simplify the numerator:
[tex]\[
4(x + 3) = 4x + 12
\][/tex]
So, it becomes:
[tex]\[
\frac{4x + 12}{(x - 5)(x + 3)(x + 7)}
\][/tex]
4. Subtract the Numerators:
- Now we subtract the two fractions with the common denominator:
[tex]\[
\frac{x^2 + 7x}{(x - 5)(x + 3)(x + 7)} - \frac{4x + 12}{(x - 5)(x + 3)(x + 7)} = \frac{(x^2 + 7x) - (4x + 12)}{(x - 5)(x + 3)(x + 7)}
\][/tex]
5. Combine and Simplify the Numerator:
- Combine the numerators:
[tex]\[
(x^2 + 7x) - (4x + 12) = x^2 + 7x - 4x - 12 = x^2 + 3x - 12
\][/tex]
6. Write the Final Result:
- The difference between the two fractions is:
[tex]\[
\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}
\][/tex]
So the simplified difference is
[tex]\[
\frac{x^2 + 3x - 12}{(x - 5)(x + 3)(x + 7)}
\][/tex]
