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

2 min read Jun 17, 2024
(x-5)^4=(x-5)^6

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.

Understanding the Equation

  • 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.

Solving the Equation

  1. 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
  2. Factoring: We can factor out a common factor of (x-5)^4:

    • (x-5)^4 [1 - (x-5)^2] = 0
  3. 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
  4. 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

Solution

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

  • x = 5
  • x = 6
  • x = 4

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