(x-5)2-9=0

Jasonbradley

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Jun 17, 2024

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Solving the Quadratic Equation: (x-5)² - 9 = 0

This article will guide you through the process of solving the quadratic equation (x-5)² - 9 = 0.

Understanding the Equation

The equation (x-5)² - 9 = 0 is a quadratic equation because it has a term with x² in it. Quadratic equations can be solved using several methods, and we will explore two common techniques:

1. Factoring

• Step 1: Expand the square: (x-5)² expands to (x-5)(x-5) = x² - 10x + 25.
  This gives us: x² - 10x + 25 - 9 = 0
• Step 2: Simplify the equation: Combine the constant terms: x² - 10x + 16 = 0
• Step 3: Factor the quadratic expression: The expression factors into (x-8)(x-2) = 0.
• Step 4: Solve for x: For the product of two factors to equal zero, at least one of the factors must be zero. Therefore, either:
  • x - 8 = 0, which means x = 8
  • x - 2 = 0, which means x = 2

2. Using the Square Root Property

• Step 1: Isolate the squared term: Add 9 to both sides of the equation: (x-5)² = 9
• Step 2: Take the square root of both sides: √(x-5)² = ±√9
• Step 3: Solve for x: x - 5 = ±3
  • x = 5 + 3 = 8
  • x = 5 - 3 = 2

Conclusion

Both methods lead to the same solutions: x = 8 and x = 2. The best method to use depends on your preference and the specific equation. Factoring is generally quicker if the equation can be easily factored, while the square root property is a reliable approach for more complex equations.