Problem 21 - Algebra and Trigonometry

Question

\(21-34\) . Perform the multiplication or division and simplify. $$ \frac{4 x}{x^{2}-4} \cdot \frac{x+2}{16 x} $$

Step-by-Step Solution

Answer

\(\frac{1}{4(x-2)}\).

Step 1: Factor the Denominator

First, factor the quadratic expression in the denominator of the first fraction: \(x^2-4\). This is a difference of squares, which can be factored as \((x-2)(x+2)\). So, \(\frac{4x}{x^2-4}\) becomes \(\frac{4x}{(x-2)(x+2)}\).

Step 2: Simplify the Expression

Rewrite the expression using the factored form: \(\frac{4x}{(x-2)(x+2)} \cdot \frac{x+2}{16x}\).

Step 3: Cancel Common Factors

Look for common factors in the numerators and denominators and cancel them out: The term \(x+2\) appears in both a numerator and a denominator. Cancel \(x+2\) from both, and \(4x\) can cancel with \(16x\) partially, reducing it to \(16\div4=4\). Now the expression becomes \(\frac{1}{(x-2) \cdot 4}\).

Step 4: Simplify Further

The remaining expression is \(\frac{1}{4(x-2)}\), which is already simplified as far as possible.

Key Concepts

Factoring Quadratics

When encountering a quadratic expression in a denominator like \(x^2-4\), it's essential to know how to factor it. Factoring quadratics is a technique utilized to simplify expressions and solve equations efficiently. A quadratic expression is typically in the form \(ax^2 + bx + c\). In the case of \(x^2 - 4\), we notice that this is a special kind of quadratic known as the "difference of squares." The difference of squares is characterized by two terms that are perfect squares separated by a subtraction sign. Recognizing these allows us to factor quickly.

• The expression \(x^2 - 4\) can be written as \((x)^2 - (2)^2\).
• Using the difference of squares formula, \(a^2 - b^2 = (a - b)(a + b)\), we factor it as \((x-2)(x+2)\).

This step transforms the problem and makes it easier to simplify later on.

Simplifying Fractions

Once you have the quadratics factored, the next step is simplifying the fraction itself. Simplifying fractions involves identifying and cancelling out common terms in the numerator and denominator.

• For the expression \(\frac{4x}{(x-2)(x+2)} \cdot \frac{x+2}{16x}\), notice that \(x+2\) appears in both a numerator and a denominator.
• Cancel this common factor, as terms that appear in both positions can be simplified away, reducing the complexity.
• Similarly, with the terms \(4x\) and \(16x\), cancel out the common factor \(4x\), resulting in a further simplification to \(\frac{1}{4(x-2)}\).

Being able to simplify fractions effectively is crucial as it reduces expressions to their simplest form, making calculations and problem-solving much easier.

Difference of Squares

The concept of a "difference of squares" is particularly handy when working with algebraic expressions. It arises when two squared terms are subtracted, which presents an opportunity for quick factoring.

• A generic difference of squares is written as \(a^2 - b^2\), and it can be factored into \((a - b)(a + b)\).
• This principle applies directly to expressions like \(x^2 - 4\), where you recognize \(x^2\) as \((x)^2\) and \(4\) as \((2)^2\).

Using the difference of squares rule is a powerful shortcut in algebra that simplifies the factoring process, helping you tackle complex expressions more effectively.