3 min read Jun 17, 2024

Solving the Equation: (x^2-4x)^2 + (x-2)^2 = 10

This equation presents a challenge due to its high degree and complex structure. Let's break down the steps to find its solutions.

1. Simplifying the Equation

The first step is to simplify the equation by expanding the squares:

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

Substituting these back into the original equation, we get:

x^4 - 8x^3 + 16x^2 + x^2 - 4x + 4 = 10

Combining like terms, we obtain:

x^4 - 8x^3 + 17x^2 - 4x - 6 = 0

2. Finding Solutions

Unfortunately, this equation is a quartic equation (degree 4) and does not have a straightforward general formula for finding solutions like quadratic equations do.

We can use a combination of techniques to find solutions:

  • Factoring: In this case, factoring the quartic equation directly is challenging.
  • Numerical Methods: We can use numerical methods like Newton-Raphson method or bisection method to approximate the solutions.
  • Graphical Methods: We can plot the function y = x^4 - 8x^3 + 17x^2 - 4x - 6 and observe where the graph intersects the x-axis.

3. Exploring the Solutions

It's important to note that quartic equations can have up to four distinct solutions (real or complex). The numerical or graphical methods mentioned above can help us determine the approximate values of these solutions.

4. Conclusion

Solving the equation (x^2-4x)^2+(x-2)^2=10 involves simplifying, identifying the equation type, and utilizing appropriate techniques like numerical methods or graphical analysis to approximate the solutions. While a direct algebraic solution might not be readily available, these methods provide ways to find the values of x that satisfy the equation.

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