Question

subtract the rational expressions to find the difference. x^2-2/4x-7 - x+4/4x-7

Answer 1

To subtract the given rational expressions, we need to ensure they have a common denominator. The expressions you provided are:

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

Since both fractions already have the same denominator \(4x - 7\), we can directly subtract the numerators:

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

Now, simplify the numerator:

\[
x^2 - 2 - x - 4 = x^2 - x - 6
\]

So, we can write the expression as:

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

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

To factor \(x^2 - x - 6\), we look for two numbers that multiply to \(-6\) and add to \(-1\). The numbers \(-3\) and \(2\) fit this.

Thus, we factor it as:

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

Now, substituting back, we obtain:

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

The final result of the subtraction is:

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

This is the difference of the two rational expressions.