(x-5)^2=(x-8)^2

2 min read Jun 17, 2024

Solving the Equation: (x-5)² = (x-8)²

This equation presents a unique opportunity to explore different methods of solving quadratic equations. Here's a breakdown of how to approach it:

1. Expanding and Simplifying:

Expand both sides: (x-5)² = (x-5)(x-5) = x² - 10x + 25 (x-8)² = (x-8)(x-8) = x² - 16x + 64
Set the equation to zero: x² - 10x + 25 = x² - 16x + 64 6x = 39
Solve for x: x = 39/6 = 6.5

2. Utilizing Square Roots:

Take the square root of both sides: √(x-5)² = ±√(x-8)²
Simplify: x - 5 = ±(x - 8)
Solve for both positive and negative cases: Case 1: x - 5 = x - 8 This equation has no solution. Case 2: x - 5 = -(x - 8) 2x = 13 x = 6.5

Explanation:

The equation (x-5)² = (x-8)² implies that the expressions inside the parentheses have the same absolute value. This is true when either both expressions are equal or when one is the negative of the other. The expansion method directly tackles this while the square root approach emphasizes the concept of absolute value.

Key Takeaways:

This equation demonstrates that quadratic equations can have multiple solutions or no solutions at all.
Understanding the properties of squares and square roots is crucial for solving such equations efficiently.
Both expansion and square root methods are valid and lead to the same solution in this case.