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(2x-3)^8=(2x-3)^6

3 min read Jun 16, 2024
(2x-3)^8=(2x-3)^6

Solving the Equation: (2x-3)^8 = (2x-3)^6

This equation presents a unique opportunity to leverage the properties of exponents. Let's break down the solution step-by-step:

1. Simplifying the Equation

  • We can start by subtracting (2x-3)^6 from both sides of the equation, giving us: (2x-3)^8 - (2x-3)^6 = 0

2. Factoring out a Common Factor

  • Observe that both terms have a common factor of (2x-3)^6. We can factor this out: (2x-3)^6 * [(2x-3)^2 - 1] = 0

3. Applying the Difference of Squares

  • The expression inside the brackets is a classic example of the difference of squares: a^2 - b^2 = (a + b)(a - b). Applying this, we get: (2x-3)^6 * [(2x-3) + 1][(2x-3) - 1] = 0

4. Simplifying Further

  • Simplifying the expressions within the brackets: (2x-3)^6 * (2x-2)(2x-4) = 0

5. Finding the Solutions

  • Now, for the entire product to be equal to zero, at least one of the factors must be zero. This gives us three potential solutions:
    • (2x-3)^6 = 0: This leads to 2x-3 = 0, resulting in x = 3/2.
    • 2x-2 = 0: This leads to x = 1.
    • 2x-4 = 0: This leads to x = 2.

6. Verifying the Solutions

  • It's always a good practice to verify your solutions by substituting them back into the original equation. In this case, all three solutions satisfy the equation.

Conclusion

Therefore, the solutions to the equation (2x-3)^8 = (2x-3)^6 are:

  • x = 3/2
  • x = 1
  • x = 2

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