(3x+2)(x^2-1)=(9x^2-4)(x+1)

Jasonbradley

3 min read Jun 16, 2024

Solving the Equation: (3x+2)(x^2-1) = (9x^2-4)(x+1)

This article will guide you through the process of solving the equation (3x+2)(x^2-1) = (9x^2-4)(x+1).

Expanding Both Sides

First, we need to expand both sides of the equation by using the distributive property (or FOIL method):

Left Side: (3x+2)(x^2-1) = 3x(x^2-1) + 2(x^2-1) = 3x^3 - 3x + 2x^2 - 2
Right Side: (9x^2-4)(x+1) = 9x^2(x+1) - 4(x+1) = 9x^3 + 9x^2 - 4x - 4

Now, our equation looks like this: 3x^3 - 3x + 2x^2 - 2 = 9x^3 + 9x^2 - 4x - 4

Rearranging the Equation

To solve for x, we need to rearrange the equation so that all terms are on one side and the other side is equal to zero. Let's move all terms to the left side:

3x^3 - 3x + 2x^2 - 2 - (9x^3 + 9x^2 - 4x - 4) = 0

Simplifying the equation:

3x^3 - 3x + 2x^2 - 2 - 9x^3 - 9x^2 + 4x + 4 = 0

-6x^3 - 7x^2 + x + 2 = 0

Finding the Solutions

Now, we have a cubic equation. Finding solutions for cubic equations can be challenging and often involves advanced techniques like factoring or using the Rational Root Theorem.

In this case, the equation can be factored as follows:

Factor out a -1: (6x^3 + 7x^2 - x - 2) = 0
Factor by grouping: (6x^3 + 7x^2) - (x + 2) = 0
Factor out common terms: x^2(6x + 7) - 1(6x + 7) = 0
Final factorization: (x^2 - 1)(6x + 7) = 0

Now, we have two factors that equal zero. This means either (x^2 - 1) = 0 or (6x + 7) = 0.

Solving for x:

x^2 - 1 = 0 => x^2 = 1 => x = ±1
6x + 7 = 0 => 6x = -7 => x = -7/6

Conclusion

Therefore, the solutions to the equation (3x+2)(x^2-1) = (9x^2-4)(x+1) are:

x = 1, x = -1, and x = -7/6