Simplify:
[tex]\begin{array}{c}
\frac{x^2-4}{x+2}-\frac{x^2+3}{x-2} \\
\frac{[?] x+}{x-}\end{array}[/tex]
Answer (1)
Factor the numerator of the first term: x 2 − 4 = ( x − 2 ) ( x + 2 ) .
Simplify the first term: x + 2 x 2 − 4 = x − 2 .
Rewrite the expression with a common denominator: x − 2 ( x − 2 ) ( x − 2 ) − ( x 2 + 3 ) .
Simplify the expression to obtain the final result: x − 2 − 4 x + 1 .
Explanation
Understanding the Problem We are asked to simplify the expression x + 2 x 2 − 4 − x − 2 x 2 + 3 and express the result in the form x − C A x + B , where A , B , and C are constants.
Factoring the Numerator First, we factor the numerator of the first term: x 2 − 4 = ( x − 2 ) ( x + 2 ) .
Simplifying the First Term Then we simplify the first term: x + 2 x 2 − 4 = x + 2 ( x − 2 ) ( x + 2 ) = x − 2.
Rewriting the Expression Now we rewrite the expression as ( x − 2 ) − x − 2 x 2 + 3 .
Finding a Common Denominator To combine the terms, we find a common denominator: x − 2 ( x − 2 ) ( x − 2 ) − ( x 2 + 3 ) .
Expanding the Numerator Next, we expand the numerator: x − 2 x 2 − 4 x + 4 − x 2 − 3 .
Simplifying the Numerator Finally, we simplify the numerator: x − 2 − 4 x + 1 .
Final Result The simplified expression is x − 2 − 4 x + 1 .
Examples
Simplifying rational expressions is a fundamental skill in algebra, useful in various real-world applications. For instance, when designing structures, engineers often encounter complex equations involving rational functions. Simplifying these expressions allows for easier analysis and optimization of the design, ensuring stability and efficiency. Similarly, in economics, simplifying rational expressions can aid in modeling and understanding complex market behaviors, leading to better decision-making and resource allocation.
