Problem 77 - Prealgebra and Introductory Algebra

Question

Simplify. $$\frac{x+4}{x^{2}-x-42}+\frac{3}{7-x}$$

Step-by-Step Solution

Answer

The simplified form of the given expression is \(\frac{-2x-14}{(x-7)(x+6)}\).\n Note that \(x \neq 7\) and \(x \neq -6\) since these values will make the denominator equal zero which is undefined in mathematics.

Step 1: Factor the quadratic equation

Factor the quadratic equation in the first fraction's denominator. The quadratic equation \(x^{2}-x-42\) factors into \((x-7)(x+6)\). So, the first fraction can be rewritten as \(\frac{x+4}{(x-7)(x+6)}\).

Step 2: Adjust the signs in the second fraction's denominator

Note that \(7-x\) is the same as \(-(x-7)\). So, the second fraction can be written as \(\frac{-3}{(x-7)}\).

Step 3: Make the denominators the same

To add or subtract fractions, they must have the same denominator. Multiply the second fraction by \(\frac{(x+6)}{(x+6)}\) to make the denominators the same. This results in \(\frac{-3(x+6)}{(x-7)(x+6)}\).

Step 4: Combine the fractions

Now we can combine the area fractions as they have the same denominators. This results in \(\frac{(x+4)-3(x+6)}{(x-7)(x+6)}\).

Step 5: Simplify the fraction

Simplify the numerator by expanding -3(x+6) and combining like terms. This gives result \(\frac{-2x-14}{(x-7)(x+6)}\).

Key Concepts

Factoring Quadratics

A quadratic expression like \( x^2 - x - 42 \) can often be factored into a product of binomials. Factoring quadratics is a key step because it allows us to break down complex expressions into simpler parts. To factor a quadratic, we look for two numbers that multiply to give the constant term (in this case, -42), and add to give the middle coefficient (in this case, -1).

In this exercise, the quadratic \( x^2 - x - 42 \) is factored as \( (x - 7)(x + 6) \). This tells us that if you expand \( (x - 7)(x + 6) \) using the distributive property (often called FOIL for binomials), you'll get back to the original expression. Factoring is crucial here, as it sets up the fractions to have a common denominator later.

Remember to always check your work by multiplying out your factors to ensure they return to the original quadratic.

Common Denominator

When adding or subtracting algebraic fractions, having a common denominator is essential. It allows you to combine the fractions into one expression. In our example, after factoring \( x - 7 \) and adjusting to \( -(x - 7) \), we need both fractions to have the same base: \( (x-7)(x+6) \).

To achieve a common denominator in our example problem, we multiply the second fraction \( \frac{-3}{x-7} \) by \( \frac{(x+6)}{(x+6)} \). This gives the denominator \( (x - 7)(x + 6) \) for both fractions. Although this does not change the value of the second fraction, it matches its terms to the first, which makes addition or subtraction straightforward.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form. This process requires us to combine and simplify terms wherever possible. After ensuring the fractions in our exercise have a common denominator, we combine them into one fraction: \( \frac{(x + 4) - 3(x + 6)}{(x - 7)(x + 6)} \).

Next, we distribute and simplify the numerator: \( (x + 4) - 3(x + 6) \). First distribute \( -3 \) to terms inside parentheses: \( -3 \times x \) and \( -3 \times 6 \), resulting in \( -3x - 18 \). Combine like terms with \( x + 4 \), giving \( -2x - 14 \).

The final simplified algebraic fraction expression is \( \frac{-2x - 14}{(x - 7)(x + 6)} \), which is a simpler and more condensed version of our initial problem. In simplifying, ensure you've combined correctly to avoid mistakes.