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Magazine Chapter 6: Problem 28
Perform the indicated operations and simplify. $$\frac{2}{n^{2}+4 n+4}-\frac{3}{4+2 n}$$
Short Answer
Expert verified
The simplified expression is \(\frac{-3n - 2}{(n+2)^2}\).
Step by step solution
01
Factor the Denominator of First Fraction
The denominator of the first fraction is \(n^2 + 4n + 4\). This is a perfect square trinomial and can be factored as \((n+2)^2\). Thus, the first fraction becomes \(\frac{2}{(n+2)^2}\).
02
Simplify the Denominator of Second Fraction
The denominator of the second fraction is \(4 + 2n\). Rewriting it in a standard form gives \(2n + 4\), which can be factored as \(2(n+2)\). Thus, the second fraction becomes \(\frac{3}{2(n+2)}\).
03
Find a Common Denominator
The common denominator for both fractions is \((n+2)^2\). We rewrite the second fraction to have this common denominator: \(\frac{3}{2(n+2)} = \frac{3}{n+2} \times \frac{1}{2}\). Multiply the numerator and denominator by \(n+2\) to get \(\frac{3(n+2)}{2(n+2)^2}\).
04
Combine the Fractions
The first fraction remains as \(\frac{2}{(n+2)^2}\), and the second fraction, after adjusting to the common denominator, is \(\frac{3(n+2)}{2(n+2)^2}\). Combine these: \(\frac{2}{(n+2)^2} - \frac{3(n+2)}{2(n+2)^2} = \frac{2 - \frac{3(n+2)}{2}}{(n+2)^2}\).
05
Simplify the Numerator
The numerator is \(2 - \frac{3(n+2)}{2}\). To simplify, multiply through by 2 to eliminate the fraction: \(2(2) - 3(n+2) = 4 - 3(n+2)\). Expand and simplify: \(4 - 3n - 6 = -3n - 2\).
06
Express the Final Fraction
Combine the simplified numerator with the common denominator: \(\frac{-3n - 2}{(n+2)^2}\). Thus, this is the simplified version of the combined fractions.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Trinomials
Factoring trinomials is a crucial step in simplifying algebraic fractions. Trinomials are polynomials with three terms, often of the form \(ax^2 + bx + c\). In this exercise, the denominator of the first fraction \(n^2 + 4n + 4\) is a perfect square trinomial. This means it can be expressed as a square of a binomial. In this case, it factors to \((n+2)^2\). To factor a trinomial:
- Identify if it can be written as a perfect square.
- Write it in the form \((a+b)^2\) where \(a\) and \(b\) are terms that satisfy the expanded form \(a^2 + 2ab + b^2\).
Common Denominator
When dealing with fractions, finding a common denominator is vital for addition and subtraction. In algebra, the common denominator allows you to express different fractions with the same base. For this exercise:
- The first fraction's denominator is \((n+2)^2\).
- The second fraction, after factoring, has its denominator as \(2(n+2)\).
Simplifying Fractions
Simplifying fractions is the process of making a fraction as simple as possible. This involves reducing the numerator and denominator to their simplest forms. In this example, the simplification involves several steps:
- The subtraction of two fractions already adjusted for a common denominator.
- Simplifying the expression in the numerator.
- Eliminating complex terms to reach a straightforward expression.
Algebraic Operations
Algebraic operations encompass a variety of techniques to add, subtract, multiply, or divide algebraic expressions. For fractions, specifically, understanding and executing these operations can effectively solve problems. For example:
- Addition and subtraction require a common denominator.
- Multiplication involves multiplying numerators and denominators directly.
- Division, inverse multiplication, flips the second fraction and proceeds as multiplication.
