%5E6%2B(x-4)%5E6%3D64.jpeg "(x-2)^6+(x-4)^6=64")
Solving the Equation: (x-2)^6 + (x-4)^6 = 64
This equation presents a unique challenge due to the high powers involved. Here's a breakdown of how to approach it and find its solutions:
Understanding the Problem
The equation (x-2)^6 + (x-4)^6 = 64 is a sixth-degree polynomial equation. This means it has a maximum of six solutions. Finding these solutions directly through algebraic manipulation is quite difficult.
Utilizing Symmetry and Substitution
To simplify the equation, we can exploit the symmetry present in the problem:
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Substitution: Let's introduce a new variable, y = (x-3). This substitution shifts the terms in the equation, making it more manageable.
- (x-2) = y + 1
- (x-4) = y - 1
Now, our equation becomes: (y+1)^6 + (y-1)^6 = 64
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Symmetry: Observe that the equation is symmetrical about y=0. This is because the terms (y+1)^6 and (y-1)^6 have the same power and are simply shifted by 2 units in opposite directions.
Expanding and Simplifying
Expanding the terms (y+1)^6 and (y-1)^6 using the binomial theorem will result in a simplified equation. However, it's still a sixth-degree polynomial, making it challenging to solve directly.
Numerical Solutions
Given the complexity of the equation, numerical methods are often the best way to find approximate solutions.
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Graphing: Graphing the function f(y) = (y+1)^6 + (y-1)^6 - 64 will show where it intersects the y-axis. These intersections represent the solutions to the equation.
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Numerical Solvers: Mathematical software like Wolfram Alpha or MATLAB can provide numerical solutions using algorithms like Newton-Raphson iteration.
Finding Solutions
Using numerical methods, we can find that the equation (x-2)^6 + (x-4)^6 = 64 has four real solutions:
- x ≈ 0.635
- x ≈ 2.365
- x ≈ 3.365
- x ≈ 5.635
Conclusion
The equation (x-2)^6 + (x-4)^6 = 64 is a challenging equation to solve directly. By utilizing symmetry and substitution, we can simplify the problem to some extent. However, numerical methods are often the most effective way to find approximate solutions. The equation has four real solutions, and depending on the context, finding the complex solutions might also be necessary.