(3/4)^6x+10-x^2 27/64

2 min read Jun 16, 2024
(3/4)^6x+10-x^2 27/64

Solving the Equation: (3/4)^(6x+10-x^2) = 27/64

This problem involves simplifying and solving an equation with an exponential term. Let's break it down step by step:

1. Expressing Both Sides with the Same Base

The key to solving this equation is to express both sides with the same base. Notice that:

  • 27/64 can be written as (3/4)³.

Now, our equation becomes:

(3/4)^(6x+10-x²) = (3/4)³

2. Equating Exponents

Since the bases are the same, we can equate the exponents:

6x + 10 - x² = 3

3. Rearranging and Solving the Quadratic Equation

Rearrange the equation to get a standard quadratic form:

x² - 6x - 7 = 0

This quadratic equation can be factored:

(x - 7)(x + 1) = 0

Therefore, the solutions for x are:

  • x = 7
  • x = -1

Solution Verification

It's always a good practice to substitute the obtained solutions back into the original equation to verify their validity.

For x = 7:

(3/4)^(6(7)+10-(7)²) = (3/4)³

(3/4)³ = (3/4)³ (This verifies the solution)

For x = -1:

(3/4)^(6(-1)+10-(-1)²) = (3/4)³

(3/4)³ = (3/4)³ (This also verifies the solution)

Conclusion

The solutions to the equation (3/4)^(6x+10-x²) = 27/64 are x = 7 and x = -1. Both solutions are valid and satisfy the original equation.