Problem 65 - Algebra 2

Question

Simplify each expression. \(\frac{1}{x^{2}+6 x+8}+\frac{3}{x+4}\)

Step-by-Step Solution

Answer

\(\frac{3x + 7}{(x+2)(x+4)}\).

Step 1: Factor the Denominators

The first denominator is \[ x^2 + 6x + 8. \]To factor it, look for two numbers that multiply to 8 and add to 6. These numbers are 2 and 4, so the factorization is \[(x+2)(x+4). \]The second term is already expressed as \(\frac{3}{x+4}\).

Step 2: Find a Common Denominator

The common denominator for \( \frac{1}{(x+2)(x+4)} \) and \( \frac{3}{x+4} \) is \((x+2)(x+4)\). The second fraction, \( \frac{3}{x+4} \), needs to be adjusted to have this common denominator.

Step 3: Adjust the Second Term

To adjust \( \frac{3}{x+4} \) so it has the common denominator \((x+2)(x+4)\), multiply both the numerator and the denominator by \((x+2)\):\[\frac{3}{x+4} = \frac{3(x+2)}{(x+2)(x+4)} = \frac{3x + 6}{(x+2)(x+4)}.\]

Step 4: Combine the Fractions

Now you can add the fractions:\[\frac{1}{(x+2)(x+4)} + \frac{3x + 6}{(x+2)(x+4)} = \frac{1 + 3x + 6}{(x+2)(x+4)}.\]

Step 5: Simplify the Numerator

Combine like terms in the numerator:\[1 + 3x + 6 = 3x + 7.\]So the expression becomes:\[\frac{3x + 7}{(x+2)(x+4)}.\]

Key Concepts

Factoring Quadratics

When tackling the problem of simplifying algebraic expressions, one useful method is factoring quadratics. Quadratics are expressions of the form \( ax^2 + bx + c \). The key to factoring these is to find two numbers that both multiply to the constant term \(c\) and add to the coefficient of the middle term \(b\).
To factor \(x^2 + 6x + 8\), we search for two numbers that multiply to 8 and sum up to 6. These numbers are 2 and 4. Therefore, we can rewrite the expression as \((x+2)(x+4)\).
This step transforms the quadratic into a product of two binomials, which is crucial for simplifying rational expressions.

Finding Common Denominator

Another critical step in simplifying expressions is finding a common denominator, especially when you're working with rational expressions. A common denominator allows you to combine fractions easily.
In our example, we had two fractions: \(\frac{1}{x^2 + 6x + 8}\) and \(\frac{3}{x + 4}\). After factoring \(x^2 + 6x + 8\) to \((x+2)(x+4)\), the common denominator for these expressions is \((x+2)(x+4)\).
By establishing this common base, we can prepare to combine the fractions more effortlessly in the subsequent steps.

Combining Rational Expressions

Once a common denominator has been identified, the next step is to combine the rational expressions into a single fraction. This involves adjusting each fraction to ensure it has the same denominator.
For \(\frac{3}{x+4}\), multiply both the numerator and the denominator by \((x+2)\) to get \(\frac{3(x+2)}{(x+2)(x+4)}\). This makes the denominators identical: \((x+2)(x+4)\).

• The expression \(\frac{1}{(x+2)(x+4)}\) remains the same since it already matches the common denominator.
• Now, you can combine the fractions: \(\frac{1 + 3(x+2)}{(x+2)(x+4)}\).
• Bring them together under one fraction by adding the numerators: \(\frac{1 + 3x + 6}{(x+2)(x+4)}\).

Simplifying Numerators

Finally, the last step of simplification is to address the numerator of your combined fraction. This often involves simplifying the expression by combining like terms to get it into its simplest form.
In our problem, after combining numerators, we have \(1 + 3x + 6\). By collecting like terms, this simplifies to \(3x + 7\).
Thus, the final simplified expression becomes \(\frac{3x + 7}{(x+2)(x+4)}\). Simplifying numerators is essential as it ensures the expression is in its most condensed and readable form, making it easier to understand and use in further calculations.