%5Ex2%2B1-%3D-(x-3)%5E3x%2B11.jpeg "(x-3)^x2+1 = (x-3)^3x+11")
Solving the Equation: (x-3)^x2+1 = (x-3)^3x+11
Introduction
In this article, we will explore the solution to the equation (x-3)^x2+1 = (x-3)^3x+11. This equation involves exponential functions and requires careful manipulation to solve.
Simplifying the Equation
Let's start by rewriting the equation:
(x-3)^x2+1 = (x-3)^3x+11
To simplify the equation, we can start by rewriting the exponential functions in a more compact form:
(x-3)^(2x+1) = (x-3)^(3x+11)
Equality of Exponential Functions
Since the bases of the exponential functions are the same, we can equate the exponents:
2x+1 = 3x+11
Solving for x
Now, we can solve for x by subtracting 2x from both sides of the equation:
x+1 = 11
Subtracting 1 from both sides gives us:
x = 10
Verifying the Solution
To verify that x=10 is indeed the solution, we can plug it back into the original equation:
(10-3)^(2(10)+1) = (10-3)^(3(10)+11)
Simplifying the equation, we get:
7^21 = 7^31
Which is true.
Conclusion
In conclusion, the solution to the equation (x-3)^x2+1 = (x-3)^3x+11 is x=10. This solution was obtained by simplifying the exponential functions and equating the exponents.
