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

2 min read Jun 17, 2024

Solving the Equation: (x^2-1)(x+2)(x-3) = (x-1)(x^2-4)(x+5)

This equation presents a challenge in its current form, but we can solve it by systematically simplifying and manipulating it. Here's a step-by-step breakdown:

1. Factorization

Factor the differences of squares:
- (x^2 - 1) = (x + 1)(x - 1)
- (x^2 - 4) = (x + 2)(x - 2)
Rewrite the equation with the factored terms: (x + 1)(x - 1)(x + 2)(x - 3) = (x - 1)(x + 2)(x - 2)(x + 5)

2. Simplifying

Cancel out common factors on both sides: Notice that (x - 1), (x + 2) are present on both sides. Canceling these out, we get: (x - 3) = (x - 2)(x + 5)

3. Expanding and Solving

Expand the right side: (x - 3) = x^2 + 3x - 10
Rearrange to form a quadratic equation: 0 = x^2 + 2x - 7
Solve the quadratic equation: This can be done using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a Where a = 1, b = 2, and c = -7

x = (-2 ± √(2^2 - 4 * 1 * -7)) / 2 * 1 x = (-2 ± √(32)) / 2 x = (-2 ± 4√2) / 2 x = -1 ± 2√2

Solution

Therefore, the solutions to the equation (x^2-1)(x+2)(x-3)=(x-1)(x^2-4)(x+5) are:

x = -1 + 2√2
x = -1 - 2√2