- Factor the quadratic denominator as \((x-2)(x+9)\) to find a common denominator.
- Combine the fractions into one: \(\frac{2(x+9) + x(x-2) - (x+20)}{(x-2)(x+9)}\).
- Simplify the numerator to \(x^2-x-2\), which factors to \((x-2)(x+1)\).
- Cancel the common factor \((x-2)\) to reach the final result: \(\boxed{\frac{x+1}{x+9}}\).
Explanation
- Factor the Denominator
To simplify the expression \(\frac{2}{x-2} + \frac{x}{x+9} - \frac{x+20}{x^2+7x-18}\), we first factor the quadratic denominator in the third term. We look for two numbers that multiply to \(-18\) and add to \(7\). These numbers are \(9\) and \(-2\). Therefore, we have \(x^2 + 7x - 18 = (x+9)(x-2)\). The expression is defined for all \(x\) except \(x=2\) and \(x=-9\).
- Rewrite with a Common Denominator
With the common denominator identified as \((x-2)(x+9)\), we rewrite each term to have this denominator:
\[ \frac{2(x+9)}{(x-2)(x+9)} + \frac{x(x-2)}{(x-2)(x+9)} - \frac{x+20}{(x-2)(x+9)} \]
Combining the numerators, we get:
\[ \frac{2(x+9) + x(x-2) - (x+20)}{(x-2)(x+9)} \]
- Simplify the Numerator
Now, we expand the terms in the numerator:
\[ 2x + 18 + x^2 - 2x - x - 20 \]
Grouping the like terms, we have:
\[ x^2 + (2x - 2x - x) + (18 - 20) = x^2 - x - 2 \]
So the expression becomes:
\[ \frac{x^2 - x - 2}{(x-2)(x+9)} \]
- Final Simplification
Finally, we factor the numerator \(x^2 - x - 2\). We look for two numbers that multiply to \(-2\) and add to \(-1\). These are \(-2\) and \(1\), so \(x^2 - x - 2 = (x-2)(x+1)\). Substituting this into our expression:
\[ \frac{(x-2)(x+1)}{(x-2)(x+9)} \]
Canceling the common factor \((x-2)\) (assuming \(x
eq 2\)), we obtain the simplified result:
\[ \frac{x+1}{x+9} \]
Comparing this result to the given options, we find it corresponds to the second option.
Examples
Simplifying rational expressions is a fundamental skill in algebra, commonly used in engineering and physics when calculating complex circuits or motion equations. For example, if you are designing a filter circuit, you might arrive at a complex expression representing the gain of the circuit. Simplifying this allows you to determine the frequency response or the stability of the circuit more easily, as seen in this exercise where a complex fraction is reduced to a simple ratio.