Problem 52 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{2}{x^{2}+4 x+4}+\frac{1}{x+2}\)

Step-by-Step Solution

Answer

\(\frac{x + 4}{(x+2)^2}\)

Step 1: Factor the Denominator

The expression has two denominators: \(x^2 + 4x + 4\) and \(x + 2\). Begin by factoring the quadratic \(x^2 + 4x + 4\).\[x^2 + 4x + 4 = (x+2)^2\]

Step 2: Identify a Common Denominator

We need a common denominator to add the fractions. The factored form of the first denominator is \((x+2)^2\), and the second denominator is \(x+2\). The least common denominator is \((x+2)^2\).

Step 3: Adjust the Numerators

Rewrite each fraction with the common denominator \((x+2)^2\).- The first fraction is already \(\frac{2}{(x+2)^2}\).- For the second fraction \(\frac{1}{x+2}\), multiply the numerator and denominator by \(x+2\) to get \(\frac{(x+2)}{(x+2)^2}\).

Step 4: Add the Fractions

Now add the adjusted fractions:\[\frac{2}{(x+2)^2} + \frac{x+2}{(x+2)^2} = \frac{2 + (x+2)}{(x+2)^2}\]

Step 5: Simplify the Numerator

Combine like terms in the numerator:\[2 + (x + 2) = x + 4\]

Step 6: Write the Simplified Expression

The expression simplifies to:\[\frac{x + 4}{(x+2)^2}\]

Key Concepts

Factoring

Factoring is an essential mathematical skill that involves breaking down expressions into simpler components called factors. In this context, you may often encounter quadratic expressions, such as polynomials of the form \(ax^2 + bx + c\). These polynomials can often be factored into two binomials.

In the given exercise, the denominator \(x^2 + 4x + 4\) is a quadratic expression. To factor it, look for two numbers that add up to give you the middle coefficient (in this case, 4), and multiply to give the last term (also 4). Here, the factorization is straightforward: \(x^2 + 4x + 4\) becomes

• \((x+2)(x+2)\)
• or simply \((x+2)^2\)

Recognizing these patterns makes it easier to work with complex expressions. Factoring is like "unlocking" expressions, revealing simpler forms. This is the first, crucial step in solving many algebraic problems, such as the one at hand.

Simplification

Simplification in mathematics refers to the process of making an expression as simple as possible. This could mean reducing a fraction, combining like terms, or organizing an expression.

Consider the fractions in our exercise. Initially, it's necessary to simplify each fraction to use a common denominator. By achieving this, the expression becomes easier to manipulate and understand.

Once you have rewritten the fractions with a common denominator, you can start to combine and reduce the terms. For the exercise given, the simplification process leads us to

• Combine like terms in the numerator: \(2 + (x + 2)\)
• Resulting in \(x + 4\)

The expression then becomes noticeably simpler, written as: \(\frac{x + 4}{(x+2)^2}\). This functionally equivalent expression offers a more straightforward view, helping to focus on its core mathematical behavior.

Quadratic Expressions

Quadratic expressions are polynomials that take the form \(ax^2 + bx + c\). These equations are common and feature prominently across various mathematical and scientific disciplines, showing up in everything from projectile motion to economics.

The quadratic expression sits at the core of our example, \(x^2 + 4x + 4\). If you can factor these expressions, you can significantly ease the computations in algebraic problems. Factoring transforms these expressions into a product of linear factors, simplifying operations like addition, subtraction, multiplication, or even finding roots.

In our exercise, identifying that \(x^2 + 4x + 4\) can be equated to \((x+2)^2\) not only simplifies the task but also uncovers potential solutions or important properties of the function. This makes handling quadratic equations less daunting and more methodical.

Fractions

Fractions are expressions representing parts of a whole and commonly appear in rational expressions. Adding and simplifying fractions is a core arithmetic skill, particularly crucial when solving algebraic problems like the one given.

Our exercise involves adding two rational expressions with different denominators: \(x^2 + 4x + 4\) and \(x+2\). The key to adding fractions is bringing them to a common denominator, which standardizes the fractions and allows for straightforward addition.

Steps to achieve this include:

• Factor denominators when possible
• Determine the least common denominator, in this case \((x+2)^2\)
• Adjust fractions to match this common denominator

Once fractions share a denominator, you simply add the numerators together, simplifying where possible. Understanding these fractional additions in rational expressions enhances comprehension and fluency in algebra.