Subtract the rational expressions: [tex]$\frac{a+1}{2 a}+\frac{3}{a^2}$[/tex]
A) [tex]$\frac{(a+3)(a-2)}{2 a^2}$[/tex]
B) [tex]$\frac{(a+1)(a-1)}{2 a^2}$[/tex]
Answer (2)
Find a common denominator: The least common denominator of 2 a and a 2 is 2 a 2 .
Rewrite each fraction with the common denominator: 2 a 2 ( a + 1 ) a − 2 a 2 3 ( 2 ) .
Subtract the numerators: ( a + 1 ) a − 3 ( 2 ) = a 2 + a − 6 .
Write the result as a single fraction and factor: 2 a 2 a 2 + a − 6 = 2 a 2 ( a + 3 ) ( a − 2 ) .
The final answer is 2 a 2 ( a + 3 ) ( a − 2 ) .
Explanation
Understanding the Problem We are asked to add two rational expressions: 2 a a + 1 and a 2 3 . Our goal is to simplify the sum and express it as a single rational expression.
Finding a Common Denominator To add these expressions, we need to find a common denominator. The least common denominator (LCD) of 2 a and a 2 is 2 a 2 .
Rewriting with Common Denominator Now, we rewrite each fraction with the common denominator 2 a 2 . This gives us 2 a 2 ( a + 1 ) a + 2 a 2 3 ( 2 ) which simplifies to 2 a 2 a 2 + a + 2 a 2 6 .
Adding the Numerators Next, we add the numerators: ( a 2 + a ) + 6 = a 2 + a + 6.
Combining into a Single Fraction Now we write the sum as a single fraction: 2 a 2 a 2 + a + 6 .
Checking for Factorization We try to factor the numerator, if possible. We look for two numbers that multiply to 6 and add to 1. Since there are no such integers, the numerator a 2 + a + 6 cannot be factored easily using integers.
Final Simplified Expression The final simplified expression is 2 a 2 a 2 + a + 6 .
Comparing with Given Options Now, we compare our result with the given options. We can expand the numerator in option A: 2 a 2 ( a + 3 ) ( a − 2 ) = 2 a 2 a 2 + a − 6 . This does not match our result. We can expand the numerator in option B: 2 a 2 ( a + 1 ) ( a − 1 ) = 2 a 2 a 2 − 1 . This also does not match our result. However, if we expand our simplified expression, we get: 2 a 2 a ( a + 1 ) + 6 = 2 a 2 a 2 + a + 6 .
Correcting the Operation and Finding the Match Since the provided options do not match the correct simplified expression, we should re-examine the original problem statement. The problem asks to subtract the rational expressions, but we added them. Let's correct this. The original problem is: 2 a a + 1 − a 2 3 Following the same steps as before, we find the common denominator 2 a 2 and rewrite the expression as: 2 a 2 ( a + 1 ) a − 2 a 2 3 ( 2 ) = 2 a 2 a 2 + a − 2 a 2 6 Subtracting the numerators, we get: 2 a 2 a 2 + a − 6 Now, we factor the numerator: a 2 + a − 6 = ( a + 3 ) ( a − 2 ) So the simplified expression is: 2 a 2 ( a + 3 ) ( a − 2 ) This matches option A.
Final Answer Therefore, the correct answer is A) 2 a 2 ( a + 3 ) ( a − 2 ) .
Examples
Rational expressions are used in various fields, such as physics, engineering, and economics. For example, in physics, they can be used to describe the motion of objects or the behavior of electrical circuits. In economics, they can be used to model supply and demand curves. Understanding how to add, subtract, multiply, and divide rational expressions is essential for solving problems in these fields. For instance, when analyzing the trajectory of a projectile, you might need to combine rational expressions to determine its position as a function of time. Similarly, in circuit analysis, you might use rational expressions to calculate the impedance of a complex circuit.
To subtract the expressions 2 a a + 1 − a 2 3 , first find a common denominator of 2 a 2 . After rewriting each fraction and combining them, we can factor the numerator to achieve the final expression of 2 a 2 ( a + 3 ) ( a − 2 ) , leading to option A as the answer.
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