(1-x)^4+(2-x)^4=(3-2x)^4

4 min read Jun 16, 2024

Solving the Equation: (1-x)^4 + (2-x)^4 = (3-2x)^4

This equation might look intimidating at first glance, but we can solve it using a combination of algebraic manipulation and a bit of cleverness. Let's break down the process step by step:

1. Expanding the Terms

Start by expanding the powers on both sides of the equation. This might seem tedious, but it's essential to get a clear picture of the terms involved:

(1-x)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4

(2-x)^4 = 16 - 32x + 24x^2 - 8x^3 + x^4

(3-2x)^4 = 81 - 216x + 216x^2 - 96x^3 + 16x^4

2. Simplifying the Equation

Now, substitute these expanded forms back into the original equation:

1 - 4x + 6x^2 - 4x^3 + x^4 + 16 - 32x + 24x^2 - 8x^3 + x^4 = 81 - 216x + 216x^2 - 96x^3 + 16x^4

Combine like terms:

2x^4 - 12x^3 + 30x^2 - 36x + 17 = 16x^4 - 96x^3 + 216x^2 - 216x + 81

Move all terms to one side:

14x^4 - 84x^3 + 186x^2 - 180x + 64 = 0

3. Recognizing a Pattern

Notice that all the coefficients are divisible by 2. Divide the entire equation by 2:

7x^4 - 42x^3 + 93x^2 - 90x + 32 = 0

Now, observe the coefficients. They seem to be related! If we look at the first and last terms (7 and 32), their product is 224. The product of the second and fourth terms (-42 and -90) is also 224. This pattern suggests a potential factorization.

4. Factoring the Equation

Let's try factoring by grouping:

(7x^4 - 42x^3) + (93x^2 - 90x) + (32) = 0

Factor out common factors:

7x^3(x - 6) + 9x(10x - 10) + 32 = 0

Now, we can see another potential grouping:

7x^3(x - 6) + 9x(10x - 10) + (32) = 0

Factor further:

7x^3(x - 6) + 90x(x - 1) + 32 = 0

Unfortunately, this doesn't lead to a simple factorization. We'll need to explore other methods to find the solutions.

5. Numerical Solutions

Since finding an exact algebraic solution might be challenging, we can utilize numerical methods like graphing or iterative techniques (like Newton-Raphson method) to approximate the solutions. These methods will help us identify the points where the graph of the equation intersects the x-axis, giving us the approximate values of x that satisfy the equation.

Conclusion

Solving the equation (1-x)^4 + (2-x)^4 = (3-2x)^4 involves a combination of algebraic manipulation, pattern recognition, and potentially numerical methods. While finding an exact algebraic solution might be complex, numerical techniques can provide accurate approximations for the solutions.