Problem 52 - Algebra and Trigonometry

Question

\(35-54\) . Perform the addition or subtraction and simplify. $$ \frac{x}{x^{2}-x-6}-\frac{1}{x+2}-\frac{2}{x-3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-2x - 1}{(x-3)(x+2)} \).

Step 1: Factor the Denominator

First, we need to factor the quadratic denominator \(x^2 - x - 6\). This expression factors as \((x-3)(x+2)\). So, rewrite the given expression using this factored form.

Step 2: Rewrite the Expression with a Common Denominator

Our task now is to have all terms share a common denominator of \((x-3)(x+2)\). Rewrite the given expression:\[ \frac{x}{(x-3)(x+2)} - \frac{1}{x+2} - \frac{2}{x-3} \] Adjust the denominators of \(\frac{1}{x+2}\) and \(\frac{2}{x-3}\) to match the common denominator.

Step 3: Adjust the Terms with Common Denominator

Multiply the numerator and denominator of the fractions without a common denominator, like this:\( \frac{1}{x+2} \) becomes \( \frac{1 \cdot (x-3)}{(x+2) \cdot (x-3)} \), so it's \( \frac{x-3}{(x+2)(x-3)} \).\( \frac{2}{x-3} \) becomes \( \frac{2 \cdot (x+2)}{(x-3) \cdot (x+2)} \), so it's \( \frac{2x+4}{(x-3)(x+2)} \).

Step 4: Combine the Terms

Now combine all the fractions into one single fraction with the same denominator:\[ \frac{x - (x-3) - (2x+4)}{(x-3)(x+2)} \]

Step 5: Simplify the Numerator

Simplify the numerator by removing parentheses and combining like terms:\[ x - (x-3) - (2x+4) = x - x + 3 - 2x - 4 = 3 - 2x - 4 = -2x - 1 \]

Step 6: Final Simplified Expression

Place the simplified numerator back over the common denominator:\[ \frac{-2x - 1}{(x-3)(x+2)} \]This is the simplified form of the original problem.

Key Concepts

Factoring Quadratics

When dealing with rational expressions, factoring quadratics is often the first step. Quadratic expressions follow the form \( ax^2 + bx + c \). Factoring involves rewriting this expression as a product of two binomials, such as \((x + m)(x + n)\). Understanding how to factor quadratics is essential, as it helps in simplifying expressions and solving equations.

Consider the quadratic \(x^2 - x - 6\) from the original problem. To factor it, look for two numbers that multiply to \(-6\) (the constant term) and add to \(-1\) (the coefficient of \(x\)). These numbers are \(-3\) and \(2\). So, the expression factors as \((x-3)(x+2)\).

Here are some tips to remember when factoring quadratics:

• Identify the coefficient pattern: The sum should equal the middle term \(b\) and the product should equal \(a \times c\).
• Use trial and error if the leading coefficient \(a\) is \(1\); find numbers that relate to \(b\) and \(c\).
• For more complex quadratics, try splitting the middle term or using the quadratic formula for hints.

Factoring quadratics simplifies the denominator and is crucial for the next steps in solving rational expressions.

Common Denominator

Finding a common denominator enables the combination of multiple algebraic fractions. Sharing a common denominator ensures the fractions can be added or subtracted without altering their overall value. This is especially important when simplifying expressions like in our original problem.

In our exercise, we need to make each fraction's denominator \((x-3)(x+2)\). This common denominator is the least common multiple of the existing denominators \(x - 3\) and \(x + 2\).

Here's how you can find a common denominator:

• Factor each denominator separately if needed.
• Identify all unique factors across the denominators.
• Construct a new denominator using all unique factors, raised to the highest power they appear in any of the denominators.

In the exercise:

• For \(\frac{1}{x+2}\), multiply numerator and denominator by \(x-3\) to get: \(\frac{x-3}{(x-3)(x+2)}\)
• For \(\frac{2}{x-3}\), multiply numerator and denominator by \(x+2\) to get: \(\frac{2(x+2)}{(x-3)(x+2)}\)

With this common denominator in place, you can easily combine and simplify the fractions.

Algebraic Fractions

Algebraic fractions, like regular fractions, can contain variables in their numerators, denominators, or both. Simplifying them often requires operations such as addition, subtraction, multiplication, or division.

To work efficiently with algebraic fractions, there are key steps to follow:

• Factoring: Just like in our example, it's crucial to factor any quadratics in the expressions, as this simplifies further calculations.
• Finding a common denominator: Before performing any addition or subtraction, establish a common denominator to normalize the fractions.
• Simplifying Expressions: Once combined under a single denominator, simplify the numerator by combining like terms and eliminating any possible factors common to both numerator and denominator.

In our original problem, after obtaining a common denominator, you simplify the expression:

• Combine the expressions into one numerator: \(x - (x-3) - (2x+4)\)
• Simplify by removing parentheses and combining like terms: \(-2x - 1\)

The final simplified version is \(\frac{-2x - 1}{(x-3)(x+2)}\). Handling algebraic fractions might seem intricate, but with practice, these steps become second nature.