Problem 59 - Algebra 2

Question

Simplify each expression. $$ \frac{2 x}{x^{2}-2 x-3}-\frac{7}{x^{2}-9} $$

Step-by-Step Solution

Answer

Therefore, the given expression simplifies to \( \frac{-5x - 7}{(x-3)(x+1)(x+3)} \)

Step 1: Factorization

Factorize the quadratic expressions in the denominators. The given expressions can be written as: \[\frac{2x}{(x-3)(x+1)} - \frac{7}{(x-3)(x+3)}\]

Step 2: Generating Common Denominator

The denominator of the fractions need to be identical to combine them. Thus, by multiplying the second fraction by \((x+1)/(x+1)\), we generate a common denominator: \[\frac{2x}{(x-3)(x+1)} - \frac{7(x+1)}{(x-3)(x+1)(x+3)}\]

Step 3: Combining Fractions

Now, as the denominators for the fractions are the same, we can combine them into one fraction: \[\frac{2x - 7(x+1)}{(x-3)(x+1)(x+3)}\]

Step 4: Simplifying Numerator

By simplifying the numerator we get: \[\frac{2x - 7x - 7}{(x-3)(x+1)(x+3)}\]which results in: \[\frac{-5x - 7}{(x-3)(x+1)(x+3)}\]

Key Concepts

Factoring Quadratic Expressions

Factoring quadratic expressions is like breaking down numbers into their prime factors but for polynomials. Given the example, the expressions to factor are quadratic expressions in the denominators: \(x^2 - 2x - 3\) and \(x^2 - 9\).

To factor these, we first look for two numbers that multiply to give the constant term (the last number) and add up to the middle term (the coefficient in front of \(x\)). For \(x^2 - 2x - 3\):

• Find numbers that multiply to \(-3\) and add up to \(-2\). These numbers are \(-3\) and \(+1\).
• The factored form is \((x - 3)(x + 1)\).

For \(x^2 - 9\):

• This is a difference of squares: \(x^2 - 3^2\).
• The factored form is \((x - 3)(x + 3)\).

Factoring helps simplify and is critical for operations like finding common denominators.

Common Denominators

Before you can add or subtract fractions, you must ensure they have a common denominator. This is similar to finding a shared base level when comparing quantities.

In our step-by-step solution, the denominators \((x-3)(x+1)\) and \((x-3)(x+3)\) were factored and compared. To make them common, we need both fractions to have the same bottom terms:

• The first fraction has \((x-3)(x+1)\).
• The second fraction must be transformed to include \((x+1)\), which it lacks.

By multiplying the numerator and denominator of the second fraction by \((x+1)\), we create a common denominator: \((x-3)(x+1)(x+3)\).

This allows the subtraction of the fractions to be smooth and straightforward.

Combining Fractions

After generating the common denominator, you can combine fractions easily. Combining means we turn two fractions into a single one by merging the numerators.

With common denominators, you merge the numerators as one would in this example:

• The numerators are \(2x\) and \(-7(x+1)\).
• By aligning them over the common denominator, the operation becomes straightforward: \(2x - 7(x+1)\).

Be cautious to watch your signs and ensure each part of the fractions is correctly combined. This stage sets the foundation for final simplification.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is the final step, aimed at making the expression as neat and manageable as possible.

In the example problem, after combining the fractions, the numerator becomes \(2x - 7(x+1)\). Simplifying this requires:

• Distribute \(-7\) across \(x+1\): result is \(-7x - 7\).
• Combine like terms: merge \(2x\) and \(-7x\) to get \(-5x\).

The result is \(-5x - 7\) for the numerator. This final form, \(\frac{-5x - 7}{(x-3)(x+1)(x+3)}\), is simpler and ready for interpretation or further mathematical operations if needed.

Simplification reduces complexity and clarifies the essential components of your algebraic expression.