(x-1)^2=8

2 min read Jun 17, 2024

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.