Section 4.5: Applications & Review

This section applies complex numbers and rational expressions to solve real-world problems and provides a summary review for practice.

Example 1: Real-World Application

A rectangle has an area of \( x^2 + 5x + 6 \). Factor to find its dimensions.

Step 1: Factor the quadratic: \( x^2 + 5x + 6 = (x+2)(x+3) \)

Step 2: Dimensions: length = x+2, width = x+3

Example 2: Complex Numbers Application

Solve \( (2 + 3i) - (1 - 2i) \) and interpret.

Step 1: Subtract real and imaginary parts: \( (2-1) + (3i+2i) = 1 + 5i \)

Step 2: Result: \( 1 + 5i \)

Example 3: Rational Expression Application

Simplify the expression \( \frac{x^2-4}{x^2-2x} \).

Step 1: Factor numerator and denominator: \( \frac{(x-2)(x+2)}{x(x-2)} \)

Step 2: Cancel common factor: \( \frac{x+2}{x} \)

Practice Problems

  1. Factor \( x^2 + 7x + 12 \) to find possible dimensions
  2. Simplify \( \frac{x^2+3x}{x} \)
  3. Compute \( (3 + 4i) + (2 - i) \)
  4. Solve \( \frac{x^2-9}{x+3} \)
  5. Combine \( \frac{1}{x+1} + \frac{2}{x+2} \)
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