(x+1)^2(x+2)+(x-1)^2(x-2)=12

4 min read Jun 16, 2024

Solving the Equation: (x+1)^2(x+2) + (x-1)^2(x-2) = 12

This equation presents a challenge due to the presence of squared terms and multiple factors. Let's break it down step-by-step to find the solution.

Expanding the Equation

First, we need to expand the equation by multiplying out the terms:

(x+1)^2(x+2) + (x-1)^2(x-2) = 12

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

Now the equation becomes:

(x^2 + 2x + 1)(x+2) + (x^2 - 2x + 1)(x-2) = 12

Let's multiply further:

(x^2 + 2x + 1)(x+2) = x^3 + 4x^2 + 5x + 2
(x^2 - 2x + 1)(x-2) = x^3 - 4x^2 + 5x - 2

The equation is now:

x^3 + 4x^2 + 5x + 2 + x^3 - 4x^2 + 5x - 2 = 12

Simplifying the Equation

Combining like terms, we get:

2x^3 + 10x = 12

Let's move all terms to one side:

2x^3 + 10x - 12 = 0

We can simplify this by dividing all terms by 2:

x^3 + 5x - 6 = 0

Finding the Solution

Now, we have a cubic equation. Finding the exact solution for cubic equations can be complex. There are multiple methods, including factoring, using the Rational Root Theorem, and numerical methods.

Here's how we can approach finding the solution:

Factorization: We can try to factor the equation. However, it doesn't appear to be easily factorable.
Rational Root Theorem: This theorem helps us find potential rational roots. Unfortunately, this method doesn't guarantee a solution, but it can give us possibilities to test.
Numerical Methods: Methods like the Newton-Raphson method can be used to approximate the solution numerically.

In this case, we can observe that the equation has one real root, and it lies between 1 and 2. We can use numerical methods to get a more accurate value for the solution.

Conclusion

The equation (x+1)^2(x+2) + (x-1)^2(x-2) = 12 has one real solution. While we can't find an exact analytical solution, we can use numerical methods to approximate the solution.