Problem 51 - Algebra and Trigonometry

Question

add or subtract as indicated. $$ \frac{4}{x^{2}+6 x+9}+\frac{4}{x+3} $$

Step-by-Step Solution

Answer

The result of the addition is \( \frac{4x+16}{(x+3)^{2}} \)

Step 1: Identify the common denominator

Noticing that the denominator \( x^{2}+6x+9 \) is the perfect square of the binomial \( x+3 \), which is the denominator of the second fraction. So this perfect square, \( x^{2}+6x+9 = (x+3)^{2} \), is the common denominator.

Step 2: Transform the second fraction to have the common denominator

The denominator \( x+3 \) can be rewritten as \( (x+3)^{2} \) when multiplied by \( x+3 \). So the fraction becomes \( \frac{4*(x+3)}{(x+3)^{2}} = \frac{4x+12}{(x+3)^{2}} \)

Step 3: Add fractions with the common denominator

Now both fractions have the same denominator \( (x+3)^{2} \), they can be added together: \( \frac{4x+12}{(x+3)^{2}} + \frac{4}{(x+3)^{2}} = \frac{4x+16}{(x+3)^{2}} \)

Key Concepts

Common Denominator

When dealing with algebraic fractions, a common denominator is essential for addition or subtraction. It harmonizes the fractions by bringing both under the same baseline. In our exercise, we need a common denominator to add the fractions \( \frac{4}{x^2+6x+9} \) and \( \frac{4}{x+3} \). A keen observation reveals that the denominator \( x^2+6x+9 \) is a perfect square of the binomial \( x+3 \). Thus, we can rewrite it as \((x+3)^2\). Using this insight, we understand that the common denominator for both fractions is \((x+3)^2\).

• This simplifies the process, making it easier to add the fractions.
• Recognizing perfect squares like \((x+3)^2\) helps identify common denominators quickly.

A shared denominator allows us to combine the fractions effortlessly.

Adding Fractions

Adding fractions involves bringing them under a single denominator to integrate the numerators. In this exercise, after identifying the common denominator \((x+3)^2\), we need to transform the second fraction. Initially, the fraction \(\frac{4}{x+3}\) must be adapted to share the common denominator. This requires multiplying the numerator and the denominator by \(x+3\), giving us \(\frac{4(x+3)}{(x+3)^2} = \frac{4x+12}{(x+3)^2}\).

• This step ensures both fractions have compatible structures for addition.
• Multiplying keeps the fraction value equivalent when upping the denominator.

With this setup, the fractions become \( \frac{4x+12}{(x+3)^2} \) and \( \frac{4}{(x+3)^2} \). Adding these results in \( \frac{4x+16}{(x+3)^2} \). This math relies on unified denominators to sum numerators smoothly.

Perfect Square

Perfect squares simplify many algebra problems by representing polynomials in a neat, factorable form. For instance, the polynomial \(x^2+6x+9\) is a perfect square since it can be expressed as \((x+3)^2\). Recognizing this helps streamline operations, such as finding a common denominator in fraction addition.

• Perfect squares often arise from expressions like \((a+b)^2\), where the middle term is double the product of the roots \((2ab)\).
• This characteristic aids in reducing polynomial expressions and finding factors.

In our exercise, realizing the expression as a perfect square allowed efficient handling and combination of the fractions. Appreciating these forms can greatly benefit algebraic manipulations.