(x-1)^2-4=0

Jasonbradley

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Jun 17, 2024

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Solving the Quadratic Equation: (x-1)^2 - 4 = 0

This article will guide you through the process of solving the quadratic equation (x-1)^2 - 4 = 0.

Understanding the Equation

The equation is already in a simplified form, but we can further break it down to understand its components:

• (x-1)^2: This represents the square of the binomial (x-1).
• -4: This is a constant term.
• = 0: The equation is set equal to zero, indicating we are looking for the values of x that satisfy the equation.

Solving the Equation

There are two common methods to solve this equation:

1. Using the Square Root Property:

• Isolate the squared term: Add 4 to both sides of the equation: (x-1)^2 = 4
• Take the square root of both sides: √(x-1)^2 = ±√4
• Simplify: x-1 = ±2
• Solve for x:
  • x = 2 + 1 = 3
  • x = -2 + 1 = -1

2. Expanding and Using the Quadratic Formula:

• Expand the squared term: (x-1)(x-1) - 4 = 0
• Simplify: x^2 - 2x + 1 - 4 = 0
• Combine like terms: x^2 - 2x - 3 = 0
• Apply the Quadratic Formula: x = [-b ± √(b^2 - 4ac)] / 2a Where a = 1, b = -2, and c = -3
• Substitute and solve: x = [2 ± √((-2)^2 - 4 * 1 * -3)] / 2 * 1 x = [2 ± √(16)] / 2 x = [2 ± 4] / 2
  • x = 3
  • x = -1

Conclusion

Both methods lead to the same solutions: x = 3 and x = -1. These are the two values of x that satisfy the original quadratic equation (x-1)^2 - 4 = 0.