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Solving the Equation (x-1)² = 8
This article will guide you through solving the equation (x-1)² = 8. We'll break down the steps and explore different methods to find the solutions.
Understanding the Equation
The equation involves a squared term, which indicates we're dealing with a quadratic equation. To solve it, we need to isolate the variable x.
Solving using the Square Root Property
- Take the square root of both sides: √[(x-1)²] = ±√8
- Simplify: x - 1 = ±√8
- Isolate x: x = 1 ± √8
- Simplify the radical: x = 1 ± 2√2
Therefore, the solutions to the equation (x-1)² = 8 are:
- x = 1 + 2√2
- x = 1 - 2√2
Solving by Expanding and Factoring
- Expand the square: (x-1)² = x² - 2x + 1
- Rewrite the equation: x² - 2x + 1 = 8
- Move the constant term to the left side: x² - 2x - 7 = 0
- Factor the quadratic expression: (x - 1 + 2√2)(x - 1 - 2√2) = 0
- Set each factor equal to zero and solve for x:
- x - 1 + 2√2 = 0 => x = 1 - 2√2
- x - 1 - 2√2 = 0 => x = 1 + 2√2
This method confirms the same solutions obtained using the square root property.
Conclusion
We've explored two methods for solving the equation (x-1)² = 8, demonstrating that the solutions are x = 1 + 2√2 and x = 1 - 2√2. Understanding quadratic equations and their solution techniques is essential in various mathematical applications.