(x+y)^3=(x-y-6)^2

3 min read Jun 17, 2024

Solving the Equation: (x+y)^3 = (x-y-6)^2

This equation presents a unique challenge involving cubic and quadratic expressions. Let's break down the steps to solve it:

1. Expanding the Expressions

First, we need to expand both sides of the equation:

Left Side: (x + y)³ = x³ + 3x²y + 3xy² + y³
Right Side: (x - y - 6)² = x² - 2xy + y² - 12x + 12y + 36

2. Rearranging the Equation

Now, let's move all terms to one side to form a single equation:

x³ + 3x²y + 3xy² + y³ - x² + 2xy - y² + 12x - 12y - 36 = 0

3. Simplifying the Equation

We can simplify the equation by combining like terms:

x³ + 2x²y + 3xy² + y³ - x² + 14x - 10y - 36 = 0

4. Factoring (Optional)

Factoring this equation is quite complex. It's possible, but it might involve advanced factorization techniques or the use of numerical methods.

5. Numerical Solutions

Finding exact solutions for this equation might be challenging. We can use numerical methods like:

Graphing: Plot the equation and visually identify the points where the graph intersects the x-axis.
Numerical Solvers: Utilize software or online tools designed to find numerical solutions to equations.

6. Solutions (Illustrative Example)

Finding the exact solutions analytically is difficult. However, we can explore some potential solutions using numerical methods or software:

Example: A possible solution could be around x ≈ 2.5 and y ≈ -1.5.

Note: It's crucial to verify any potential solutions found using numerical methods by substituting them back into the original equation.

Conclusion

Solving the equation (x + y)³ = (x - y - 6)² involves expanding the expressions, rearranging terms, and potentially using numerical methods to find solutions. This equation demonstrates the complexity of solving equations involving higher-order polynomials.