In Exercises 33–68, add or subtract as indicated. 4/(x²+6x+9) + 4/(x+3)

Verified step by step guidance

1. Step 1: Recognize that the denominators in the given expression are different. The first term has a denominator of \(x^2 + 6x + 9\), and the second term has a denominator of \(x + 3\). Factorize \(x^2 + 6x + 9\) to simplify the expression. Notice that \(x^2 + 6x + 9\) is a perfect square trinomial, so it can be rewritten as \((x + 3)^2\).
2. Step 2: Rewrite the expression using the factored form of the first denominator: \(\frac{4}{(x + 3)^2} + \frac{4}{x + 3}\). Now, observe that the denominators are \((x + 3)^2\) and \(x + 3\), which are not the same. To combine these fractions, we need a common denominator.
3. Step 3: Determine the least common denominator (LCD). The LCD for \((x + 3)^2\) and \(x + 3\) is \((x + 3)^2\). Rewrite the second fraction \(\frac{4}{x + 3}\) so that it has the LCD as its denominator. Multiply both the numerator and denominator of \(\frac{4}{x + 3}\) by \(x + 3\), resulting in \(\frac{4(x + 3)}{(x + 3)^2}\).
4. Step 4: Combine the two fractions under the common denominator \((x + 3)^2\). The first fraction remains \(\frac{4}{(x + 3)^2}\), and the second fraction becomes \(\frac{4(x + 3)}{(x + 3)^2}\). Add the numerators: \(4 + 4(x + 3)\), keeping the denominator \((x + 3)^2\).
5. Step 5: Simplify the numerator. Distribute the \(4\) in \(4(x + 3)\) to get \(4x + 12\). Combine this with the \(4\) from the first term to get \(4x + 16\). The final expression is \(\frac{4x + 16}{(x + 3)^2}\).

Key Concepts

Factoring Quadratics

Factoring quadratics involves rewriting a quadratic expression in the form ax^2 + bx + c as a product of two binomials. In the given expression, x^2 + 6x + 9 can be factored into (x + 3)(x + 3) or (x + 3)^2. This simplification is crucial for combining fractions with a common denominator.

Finding a Common Denominator

To add or subtract fractions, it is essential to have a common denominator. In this case, the denominators are (x^2 + 6x + 9) and (x + 3). Since (x + 3) is a factor of (x^2 + 6x + 9), the common denominator can be identified as (x + 3)^2, allowing for the fractions to be combined effectively.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator. After finding a common denominator and combining the fractions, it is important to simplify the resulting expression to make it easier to interpret and work with.