Multiple choice

The imaginary roots of the equation $\left( x ^ { 2 } + 2 \right) ^ { 2 } + 8 x ^ { 2 } = 6 x \left( x ^ { 2 } + 2 \right)$ are

  1. $1 \pm i$
  2. $2 \pm i$
  3. $-1 \pm i$
  4. $-2 \pm i$
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A Correct answer
Explanation

Let u = x^2+2. The equation is u^2 + 8x^2 = 6xu. u^2 - 6xu + 8x^2 = 0. (u-4x)(u-2x) = 0. Case 1: x^2+2 = 4x => x^2-4x+2=0 => x = (4 +/- sqrt(16-8))/2 = 2 +/- sqrt(2). Case 2: x^2+2 = 2x => x^2-2x+2=0 => x = (2 +/- sqrt(4-8))/2 = 1 +/- i. The imaginary roots are 1 +/- i.

AI explanation

Expanding the given equation (x^2 + 2)^2 + 8x^2 = 6x(x^2 + 2) yields x^4 - 6x^3 + 12x^2 - 12x + 4 = 0. This polynomial factors into two quadratics as (x^2 - 2x + 2)(x^2 - 4x + 2) = 0. The roots of x^2 - 2x + 2 = 0 are found using the quadratic formula to be 1 + i and 1 - i. The roots of x^2 - 4x + 2 = 0 are 2 + sqrt(2) and 2 - sqrt(2). Therefore, the imaginary roots are 1 + i and 1 - i.

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