$2-\frac{x+1}{x-2}-\frac{x-4}{x+2}$ can be written as a single fraction in the form $\frac{a x+b}{x^2-4}$ where $a$ and $b$ are integers. Work out the value of $a$ and the value of $b$.
Answer (1)
We are asked to simplify the expression 2 − x − 2 x + 1 − x + 2 x − 4 and express it in the form x 2 − 4 a x + b .
First, find a common denominator, which is x 2 − 4 .
Rewrite the expression with the common denominator: x 2 − 4 2 ( x 2 − 4 ) − ( x + 1 ) ( x + 2 ) − ( x − 4 ) ( x − 2 ) .
Expand and simplify the numerator: 2 x 2 − 8 − ( x 2 + 3 x + 2 ) − ( x 2 − 6 x + 8 ) = 3 x − 18 .
Thus, the expression becomes x 2 − 4 3 x − 18 , so a = 3 and b = − 18 .
The final answer is a = 3 , b = − 18 .
Explanation
Understanding the Problem We are given the expression 2 − x − 2 x + 1 − x + 2 x − 4 and we want to write it as a single fraction in the form x 2 − 4 a x + b , where a and b are integers. Our goal is to find the values of a and b .
Finding a Common Denominator First, we need to rewrite the given expression as a single fraction with a common denominator. The common denominator for the fractions is ( x − 2 ) ( x + 2 ) = x 2 − 4 . So, we rewrite each term with this common denominator:
2 = x 2 − 4 2 ( x 2 − 4 )
x − 2 x + 1 = x 2 − 4 ( x + 1 ) ( x + 2 )
x + 2 x − 4 = x 2 − 4 ( x − 4 ) ( x − 2 )
Combining the Fractions Now, we combine the fractions:
x 2 − 4 2 ( x 2 − 4 ) − ( x + 1 ) ( x + 2 ) − ( x − 4 ) ( x − 2 )
Expanding the Numerator Terms Next, we expand the terms in the numerator:
2 ( x 2 − 4 ) = 2 x 2 − 8
( x + 1 ) ( x + 2 ) = x 2 + 2 x + x + 2 = x 2 + 3 x + 2
( x − 4 ) ( x − 2 ) = x 2 − 2 x − 4 x + 8 = x 2 − 6 x + 8
Substituting the Expanded Terms Substitute these expansions back into the numerator:
2 x 2 − 8 − ( x 2 + 3 x + 2 ) − ( x 2 − 6 x + 8 )
Simplifying the Numerator Now, we simplify the numerator:
2 x 2 − 8 − x 2 − 3 x − 2 − x 2 + 6 x − 8 = ( 2 x 2 − x 2 − x 2 ) + ( − 3 x + 6 x ) + ( − 8 − 2 − 8 ) = 0 x 2 + 3 x − 18 = 3 x − 18
Writing as a Single Fraction So, the expression simplifies to:
x 2 − 4 3 x − 18
Finding a and b Comparing this to x 2 − 4 a x + b , we can see that a = 3 and b = − 18 .
Final Answer Therefore, the values of a and b are a = 3 and b = − 18 .
Examples
Understanding how to combine rational expressions is crucial in various fields, such as electrical engineering when analyzing circuits or in physics when dealing with wave interference. For instance, when calculating the total impedance of a parallel circuit with two components, you often need to combine fractions involving complex numbers. Simplifying these expressions allows engineers to determine the overall behavior of the circuit and optimize its performance. Similarly, in physics, combining wave functions often involves adding or subtracting rational expressions to understand interference patterns.
