Question

perform the indicated operations:
\\(\frac{2}{x - 2} + \frac{x}{x + 9} - \frac{x + 20}{x^2 + 7x - 18}\\)
options:
\\(\frac{22}{x^2 + 7x - 18}\\)
\\(\frac{x + 1}{x + 9}\\)
\\(\frac{x^2 - x + 2}{x^2 + 7x - 18}\\)
\\(\frac{x^2 + x + 38}{x^2 + 7x - 18}\\)

Explanation:

Step1: Factor the denominator

First, factor the quadratic denominator \(x^2 + 7x - 18\). We need two numbers that multiply to \(-18\) and add to \(7\). Those numbers are \(9\) and \(-2\), so \(x^2 + 7x - 18=(x + 9)(x - 2)\).

Step2: Find a common denominator

The denominators of the fractions are \(x - 2\), \(x + 9\), and \((x + 9)(x - 2)\). The least common denominator (LCD) is \((x + 9)(x - 2)\).

Step3: Rewrite each fraction with the LCD

• For \(\frac{2}{x - 2}\), multiply numerator and denominator by \(x + 9\): \(\frac{2(x + 9)}{(x - 2)(x + 9)}\)
• For \(\frac{x}{x + 9}\), multiply numerator and denominator by \(x - 2\): \(\frac{x(x - 2)}{(x + 9)(x - 2)}\)
• For \(\frac{x + 20}{(x + 9)(x - 2)}\), it already has the LCD.

Step4: Combine the fractions

Now, combine the numerators over the common denominator:
\[

$$\begin{align*} &\frac{2(x + 9)+x(x - 2)-(x + 20)}{(x + 9)(x - 2)}\\ =&\frac{2x + 18 + x^2 - 2x - x - 20}{(x + 9)(x - 2)}\\ =&\frac{x^2 - x - 2}{(x + 9)(x - 2)} \end{align*}$$

\]

Step5: Factor the numerator

Factor the numerator \(x^2 - x - 2\). We need two numbers that multiply to \(-2\) and add to \(-1\). Those numbers are \(-2\) and \(1\), so \(x^2 - x - 2=(x - 2)(x + 1)\).

Step6: Simplify the fraction

Now, we have \(\frac{(x - 2)(x + 1)}{(x + 9)(x - 2)}\). Cancel out the common factor \((x - 2)\) (assuming \(x
eq2\)):
\[
\frac{x + 1}{x + 9}
\]

Answer:

\(\frac{x + 1}{x + 9}\)