(x-5)^4=(x-5)^6

2 min read Jun 17, 2024

Solving the Equation (x-5)^4 = (x-5)^6

This equation presents an interesting scenario where we have the same base raised to different powers. Let's explore how to solve it.

Similar Bases: Both sides of the equation share the same base, (x-5).
Different Powers: The powers are different, 4 on the left and 6 on the right.

Rearranging: We want to bring all terms to one side to set the equation to zero.
- Subtract (x-5)^6 from both sides: (x-5)^4 - (x-5)^6 = 0
Factoring: We can factor out a common factor of (x-5)^4:
- (x-5)^4 [1 - (x-5)^2] = 0
Zero Product Property: The product of two factors is zero if and only if at least one of the factors is zero.
- Therefore, we have two possible solutions:
  - (x-5)^4 = 0
  - 1 - (x-5)^2 = 0
Solving for x:
- (x-5)^4 = 0
  - Taking the fourth root of both sides: x-5 = 0
  - Solving for x: x = 5
- 1 - (x-5)^2 = 0
  - Rearranging: (x-5)^2 = 1
  - Taking the square root of both sides: x-5 = ±1
  - Solving for x: x = 6 or x = 4

Therefore, the solutions to the equation (x-5)^4 = (x-5)^6 are: