%5E6x%2B10-x%5E2-27%2F64.jpeg "(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.