Simplify. $\frac{x^2-9 x+14}{x^2+5 x-14}$
Answer (1)
Factor the numerator: x 2 − 9 x + 14 = ( x − 2 ) ( x − 7 ) .
Factor the denominator: x 2 + 5 x − 14 = ( x + 7 ) ( x − 2 ) .
Write the rational expression with the factored numerator and denominator: ( x + 7 ) ( x − 2 ) ( x − 2 ) ( x − 7 ) .
Cancel out the common factor ( x − 2 ) to simplify the expression: x + 7 x − 7 . The final answer is x + 7 x − 7 .
Explanation
Understanding the Problem We are given the rational expression x 2 + 5 x − 14 x 2 − 9 x + 14 and asked to simplify it. To do this, we need to factor both the numerator and the denominator and then cancel any common factors.
Factoring the Numerator First, let's factor the numerator, x 2 − 9 x + 14 . We are looking for two numbers that multiply to 14 and add to -9. These numbers are -2 and -7. So, we can factor the numerator as ( x − 2 ) ( x − 7 ) .
Factoring the Denominator Next, let's factor the denominator, x 2 + 5 x − 14 . We are looking for two numbers that multiply to -14 and add to 5. These numbers are 7 and -2. So, we can factor the denominator as ( x + 7 ) ( x − 2 ) .
Simplifying the Expression Now we can write the rational expression as ( x + 7 ) ( x − 2 ) ( x − 2 ) ( x − 7 ) . We can cancel the common factor of ( x − 2 ) from the numerator and the denominator, which gives us x + 7 x − 7 .
Final Answer Therefore, the simplified expression is x + 7 x − 7 .
Examples
Rational expressions are used in various fields, such as physics and engineering, to model relationships between different quantities. For example, in electrical engineering, rational functions can describe the impedance of a circuit as a function of frequency. Simplifying these expressions allows engineers to analyze and design circuits more efficiently. Also, in physics, rational expressions can appear in formulas describing the motion of objects or the behavior of waves. Simplifying these expressions can make calculations easier and provide a clearer understanding of the underlying physical phenomena. Understanding how to simplify rational expressions is a fundamental skill that can be applied in many real-world scenarios.
