3 min read Jun 17, 2024

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

This equation looks complicated at first glance, but we can solve it by carefully expanding and simplifying. Here's how:

1. Expand the Left Side

First, we need to expand the left side of the equation:

(x² + x)² = (x² + x)(x² + x)

Using the distributive property (or FOIL method), we get:

(x² + x)² = x⁴ + 2x³ + x²

2. Rearrange the Equation

Now, let's rearrange the equation so that all terms are on one side:

x⁴ + 2x³ + x² - (x² - 2x + 1) = 0

Simplifying this, we get:

x⁴ + 2x³ + 2x - 1 = 0

3. Factorization

Unfortunately, this equation doesn't factor easily. There is no straightforward way to factor a fourth-degree polynomial like this. Therefore, we need to use more advanced techniques to find the solutions.

4. Numerical Solutions

To find the solutions, we can use numerical methods like the Rational Root Theorem or Newton-Raphson method. These methods involve iterative calculations to approximate the solutions.

Rational Root Theorem: This theorem helps us identify possible rational roots of the polynomial. We can use this information to test potential solutions and narrow down the search for the actual roots.

Newton-Raphson Method: This is an iterative method that starts with an initial guess for the root and then repeatedly refines the guess until it converges to the actual root.

5. Conclusion

Solving the equation (x² + x)² = x² - 2x + 1 involves expanding, rearranging, and utilizing numerical methods to find the solutions. While the equation doesn't factor easily, techniques like the Rational Root Theorem and Newton-Raphson method can help us find the approximate values of the roots.

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