(x^2+x+1)(x^2+x+2)-12=0

2 min read Jun 17, 2024

Solving the Equation: (x^2 + x + 1)(x^2 + x + 2) - 12 = 0

This equation presents a unique challenge with its nested quadratic expressions. We can solve it using a combination of substitution and factoring. Here's how:

1. Substitution:

Let's simplify the equation by substituting:
- y = x² + x
Now, our equation becomes:
- (y + 1)(y + 2) - 12 = 0

2. Expanding and Simplifying:

Expand the equation:
- y² + 3y + 2 - 12 = 0
Combine like terms:
- y² + 3y - 10 = 0

3. Factoring:

Factor the quadratic equation:
- (y + 5)(y - 2) = 0
Solve for y:
- y = -5 or y = 2

4. Back-Substitution:

Substitute back the original expression for y (x² + x):
- For y = -5:
  - x² + x = -5
  - x² + x + 5 = 0
- For y = 2:
  - x² + x = 2
  - x² + x - 2 = 0

5. Solving the Quadratic Equations:

Solve the quadratic equations using the quadratic formula:
- For x² + x + 5 = 0:
  - x = [-1 ± √(1² - 4 * 1 * 5)] / (2 * 1)
  - x = [-1 ± √(-19)] / 2
  - x = (-1 ± √19i) / 2 (where 'i' is the imaginary unit)
- For x² + x - 2 = 0:
  - x = [-1 ± √(1² - 4 * 1 * -2)] / (2 * 1)
  - x = [-1 ± √9] / 2
  - x = (-1 ± 3) / 2
  - x = 1 or x = -2

6. The Solutions:

Therefore, the solutions to the equation (x² + x + 1)(x² + x + 2) - 12 = 0 are:

x = (-1 + √19i) / 2
x = (-1 - √19i) / 2
x = 1
x = -2

These solutions consist of two real solutions (x = 1 and x = -2) and two complex solutions.