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(x^4+x^2+1)(x^4+x^2+2)=12

3 min read Jun 17, 2024
(x^4+x^2+1)(x^4+x^2+2)=12

Solving the Equation (x^4 + x^2 + 1)(x^4 + x^2 + 2) = 12

This problem involves solving a rather complex equation with high-degree terms. We can solve it by simplifying the equation and then using various algebraic techniques.

Simplifying the Equation

  1. Substitution: Let's make the equation easier to work with by introducing a substitution. Let y = x^2. This simplifies the equation to:

    (y^2 + y + 1)(y^2 + y + 2) = 12

  2. Expansion: Expand the left side of the equation:

    y^4 + 2y^3 + 4y^2 + 3y + 2 = 12

  3. Rearranging: Move the constant term to the left side:

    y^4 + 2y^3 + 4y^2 + 3y - 10 = 0

Solving the Quartic Equation

Now we have a quartic equation (an equation with the highest power of the variable being 4). Solving quartic equations can be quite complex, but we can try to factor it:

  1. Factoring: We can factor the equation as follows:

    (y^2 + 2y - 2)(y^2 + 5) = 0

  2. Solving the Factors: Now we have two quadratic equations to solve:

    • y^2 + 2y - 2 = 0 This equation can be solved using the quadratic formula:

      y = [-2 ± √(2^2 - 4 * 1 * -2)] / (2 * 1) y = [-2 ± √12] / 2 y = -1 ± √3

    • y^2 + 5 = 0 This equation has no real solutions, as the square of a real number cannot be negative.

Finding the Solutions for x

Remember that y = x^2. We need to substitute back and solve for x:

  1. For y = -1 + √3:

    x^2 = -1 + √3 x = ±√(-1 + √3)

  2. For y = -1 - √3:

    x^2 = -1 - √3 x = ±√(-1 - √3)

The solutions for x involve complex numbers due to the square root of negative values.

Therefore, the equation (x^4 + x^2 + 1)(x^4 + x^2 + 2) = 12 has four solutions, all of which are complex numbers.

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