Problem 57 - Algebra and Trigonometry

Question

Perform the addition or subtraction and simplify. $$ \frac{2}{x+3}-\frac{1}{x^{2}+7 x+12} $$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{2x + 7}{(x+3)(x+4)}\).

Step 1: Examine denominators

The first fraction has the denominator \(x+3\). The second fraction's denominator is \(x^2 + 7x + 12\). Notice that \(x^2 + 7x + 12\) factors into \((x+3)(x+4)\).

Step 2: Find a common denominator

The least common denominator (LCD) for the fractions \(\frac{2}{x+3}\) and \(\frac{1}{x^2 + 7x + 12}\) is \((x+3)(x+4)\).

Step 3: Adjust the first fraction

Multiply the numerator and denominator of the first fraction \(\frac{2}{x+3}\) by \(x+4\) to match the common denominator. This gives \(\frac{2(x+4)}{(x+3)(x+4)}\).

Step 4: Adjust the second fraction

The second fraction already has the common denominator, \((x+3)(x+4)\), so it remains \(\frac{1}{(x+3)(x+4)}\).

Step 5: Perform the subtraction

Now that both fractions have the same denominator, subtract the second fraction from the first: \(\frac{2(x+4)}{(x+3)(x+4)} - \frac{1}{(x+3)(x+4)} = \frac{2(x+4) - 1}{(x+3)(x+4)}\).

Step 6: Simplify the expression

Distribute the 2 in the numerator: \(2(x+4) = 2x + 8\), so the expression becomes \(\frac{2x + 8 - 1}{(x+3)(x+4)} = \frac{2x + 7}{(x+3)(x+4)}\).

Step 7: State the final result

The simplified result of the subtraction is \(\frac{2x + 7}{(x+3)(x+4)}\).

Key Concepts

Addition and Subtraction of Fractions

Understanding the addition and subtraction of fractions is crucial when dealing with rational expressions. Just as with numerical fractions, to add or subtract rational expressions, they must have a common denominator. This is because fractions represent parts of a whole, and without a common baseline, it would be like comparing apples to oranges.
To successfully add or subtract fractions:

• Ensure both fractions have the same denominator.
• If they don't have the same denominator, find a common one.
• Convert the fractions to have this common denominator.
• Perform the addition or subtraction on the numerators only, leaving the denominator unchanged.

Remember to simplify the result, if possible, by factoring or reducing common terms.

Common Denominator

Finding a common denominator is a vital step in adding and subtracting fractions, whether they are rational expressions or numerical. A common denominator allows you to put both fractions on the same footing, making them comparable.
When fractions have different denominators, the process is similar to finding the least common multiple (LCM) in arithmetic. For example, in the exercise, the denominators are \(x+3\) and \(x^2 + 7x + 12\), where the latter can be factored into \((x+3)(x+4)\).
Here's how to find a common denominator:

• Factor the denominators to find common elements.
• Identify all unique factors present in either denominator.
• Multiply these together to form the least common denominator (LCD).

In this case, the LCD is \((x+3)(x+4)\). This simplest form encompasses all parts from both fractions' denominators, making further operations straightforward.

Factoring Quadratic Expressions

Factoring is an essential algebraic technique, especially when dealing with rational expressions. It involves breaking down a complex expression into simpler components that can be multiplied together. For quadratic expressions, such as \(x^2 + 7x + 12\), factoring them allows us to identify their roots or simplify computations.
The quadratic expression \(x^2 + 7x + 12\) can be factored by finding two numbers that add to 7 (the coefficient of \(x\)) and multiply to 12 (the constant term). For this expression:

• Identify the factors of 12: 1, 2, 3, 4, 6, 12.
• Find a pair that adds up to 7: in this case, 3 and 4.
• Rewrite the expression as \((x+3)(x+4)\).

Factoring transforms \(x^2+7x+12\) into a product of two binomials, which helps in simplifying the subtraction process or further operations with rational expressions.