What is the difference?

[tex]\[
\frac{x}{x^2 - 2x - 15} - \frac{4}{x^2 + 2x - 35}
\][/tex]

A. [tex]\(\frac{x^2 + 3x + 12}{(x-3)(x-5)(x+7)}\)[/tex]

B. [tex]\(\frac{x(x + 3 - 12)}{(x + 3)(x - 5)(x + 7)}\)[/tex]

C. [tex]\(\frac{x^2 + 3x + 12}{(x + 3)(x - 5)(x + 7)}\)[/tex]

D. [tex]\(\frac{x^2 + 3x - 120}{(x + 3)(x - 5)(x + 7)}\)[/tex]



Answer :

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]

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