Evaluate the following, [tex]$\frac{a^2-b^2}{a-b}+\frac{2 a^2+b}{a+b}$[/tex] where [tex]$a=-1$[/tex] and [tex]$b=3$[/tex].
Answer (2)
Simplify the first term using difference of squares: a − b a 2 − b 2 = a + b .
Substitute a = − 1 and b = 3 into a + b to get 2 .
Substitute a = − 1 and b = 3 into a + b 2 a 2 + b to get 2 5 .
Add the two terms: 2 + 2 5 = 2 9 . The final answer is 2 9 .
Explanation
Problem Analysis We are asked to evaluate the expression a − b a 2 − b 2 + a + b 2 a 2 + b where a = − 1 and b = 3 . Let's break this down step by step.
Simplifying the First Term First, we can simplify the term a − b a 2 − b 2 . Notice that a 2 − b 2 is a difference of squares, which factors as ( a − b ) ( a + b ) . So we have: a − b a 2 − b 2 = a − b ( a − b ) ( a + b ) Since a = b , we can cancel the ( a − b ) terms, which gives us a + b .
Substituting the Values Now our expression looks like this: a + b + a + b 2 a 2 + b We are given that a = − 1 and b = 3 . Let's substitute these values into the expression.
Calculating the Terms First, let's find a + b : a + b = − 1 + 3 = 2 Next, let's find 2 a 2 + b : 2 a 2 + b = 2 ( − 1 ) 2 + 3 = 2 ( 1 ) + 3 = 2 + 3 = 5 So our expression becomes: 2 + 2 5
Final Calculation Now, let's add the terms: 2 + 2 5 = 2 4 + 2 5 = 2 9 So the final result is 2 9 .
Examples
This type of algebraic simplification and evaluation is used in many fields, such as physics and engineering, where you often need to substitute values into complex formulas to find a result. For example, you might use this in calculating the trajectory of a projectile or the stress on a structural beam. Understanding how to simplify and substitute efficiently can save time and reduce errors in these calculations.
The evaluated expression results in 2 9 . This was achieved by simplifying each term with the given values for a and b . The first term reduces to 2, and the second term is 2 5 , leading to the final sum of 2 9 .
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