Question

Subtract the rational expressions. to find the difference: x2-2/4x-7 - x+4/4x-7

Answer 1

To subtract the rational expressions \( \frac{x^2 - 2}{4x - 7} - \frac{x + 4}{4x - 7} \), we first note that both fractions have the same denominator, \( 4x - 7 \). Therefore, we can combine the numerators as follows:

\[
\frac{x^2 - 2 - (x + 4)}{4x - 7}
\]

Now we simplify the numerator:

1. Distribute the negative sign through \( x + 4 \):
\[
x^2 - 2 - x - 4
\]

2. Combine like terms:
\[
x^2 - x - 2 - 4 = x^2 - x - 6
\]

So, we now have:

\[
\frac{x^2 - x - 6}{4x - 7}
\]

Next, we factor the numerator \( x^2 - x - 6 \):

To factor \( x^2 - x - 6 \), we need two numbers that multiply to \(-6\) and add to \(-1\). These numbers are \( -3 \) and \( 2 \). Thus, we can factor it as:

\[
x^2 - x - 6 = (x - 3)(x + 2)
\]

Now we can write our expression as:

\[
\frac{(x - 3)(x + 2)}{4x - 7}
\]

Thus, the final result of the subtraction is:

\[
\frac{(x - 3)(x + 2)}{4x - 7}
\]