Problem 9 - Algebra 2

Question

Simplify each expression. $$ \frac{6}{d^{2}+4 d+4}+\frac{5}{d+2} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{5d + 16}{(d + 2)^2} \).

Step 1: Factor the Quadratic Expression

The first step is to factor the quadratic expression in the denominator of the first fraction, which is \( d^2 + 4d + 4 \). Notice that this is a perfect square trinomial, and it can be written as \((d + 2)^2\). Thus, the expression becomes \( \frac{6}{(d + 2)^2} + \frac{5}{d + 2} \).

Step 2: Find the Least Common Denominator (LCD)

The least common denominator of the fractions is \((d + 2)^2\) because it is the more extensive power of \((d + 2)\) appearing in the denominators. We will use this to rewrite each fraction with the common denominator.

Step 3: Rewrite Each Fraction with the LCD

The first fraction \( \frac{6}{(d + 2)^2} \) already has the correct denominator. For the second fraction \( \frac{5}{d + 2} \), multiply both the numerator and denominator by \(d + 2\) to obtain \( \frac{5(d + 2)}{(d + 2)^2} \).

Step 4: Simplify the Numerators

Now that both fractions have the same denominator, you can combine them: \( \frac{6}{(d + 2)^2} + \frac{5(d + 2)}{(d + 2)^2} = \frac{6 + 5(d + 2)}{(d + 2)^2} \). Simplify the numerator: \( 6 + 5(d + 2) = 6 + 5d + 10 = 5d + 16 \).

Step 5: Combine and Simplify the Expression

Combine the fractions: \( \frac{5d + 16}{(d + 2)^2} \). You cannot simplify this expression further unless there are common factors to cancel out, which in this case, there are not.

Key Concepts

Factoring Quadratics

Factoring quadratics is an essential skill in algebra that allows us to simplify expressions or solve equations. In our exercise, the quadratic expression in the denominator is \(d^2 + 4d + 4\). This particular quadratic is a perfect square trinomial. A perfect square trinomial can be rewritten as the square of a binomial, which helps in simplifying algebraic expressions.

Consider the expression \(d^2 + 4d + 4\). Here, you notice that it can be written in the form \((d + 2)^2\) because when expanded, \((d + 2)(d + 2)\) yields \(d^2 + 4d + 4\).

• Identify the square of the first term: \(d^2 = (d)^2\).
• Ensure the middle term is twice the product of the two terms being squared: \(2 \times d \times 2 = 4d\).
• Make sure the last term is the square of the second term: \(2^2 = 4\).

By recognizing such patterns, factoring quadratics becomes straightforward and will pave the way for further simplification and solving of equations.

Least Common Denominator

The least common denominator (LCD) is a vital concept when dealing with fractions, especially in algebra. It allows us to rewrite fractions in terms of a common denominator so they can be added or subtracted easily.

In the given exercise, we have two fractions: \(\frac{6}{(d + 2)^2}\) and \(\frac{5}{d + 2}\). The task is to find the LCD to combine these fractions into a single expression.

• Look at the denominators: \((d + 2)^2\) and \(d + 2\).
• Identify the common term: \(d + 2\) appears in both denominators.
• The LCD is the higher power of the common factor, which is \((d + 2)^2\) here.

With this common denominator, you can rewrite each fraction, ensuring they share the same base, which is crucial for combining and simplifying.

Combining Fractions

Combining fractions requires them to have a common denominator. Once a common denominator is found, the addition or subtraction can be performed easily. For the expressions \(\frac{6}{(d + 2)^2}\) and \(\frac{5}{d + 2}\) from our exercise, we found the least common denominator to be \((d + 2)^2\).

The first fraction \(\frac{6}{(d + 2)^2}\) already has the common denominator. The second fraction \(\frac{5}{d + 2}\) needs to be adjusted to have \((d + 2)^2\) as its denominator.

• Multiply both the numerator and denominator of the second fraction by \(d + 2\): \[ \frac{5}{d + 2} \times \frac{d + 2}{d + 2} = \frac{5(d + 2)}{(d + 2)^2} \].
• Now both fractions have the same denominator: \(\frac{6}{(d + 2)^2} + \frac{5(d + 2)}{(d + 2)^2}\).

With the denominators the same, you add the numerators, keeping the common denominator, to form one simplified fraction.

Simplifying Algebraic Expressions

Simplifying algebraic expressions means reducing them to their simplest form. After combining the fractions, the expression is \(\frac{6 + 5(d + 2)}{(d + 2)^2}\). The next step is simplifying the numerator.

Start by expanding and simplifying the terms in the numerator:

• Distribute the \(5\) in \(5(d + 2)\): \(5 \times d + 5 \times 2 = 5d + 10\).
• Combine this result with the constant \(6\):\(6 + 5d + 10 = 5d + 16\).

This leaves us with a single, simplified expression: \(\frac{5d + 16}{(d + 2)^2}\).

You need to check if further simplification is possible by finding common factors in the numerator and the denominator. In this case, \(5d + 16\) and \((d + 2)^2\) share no common factors, so the expression is already in its simplest form.