Multiple choice

The number of real roots of the equation $\displaystyle \left ( x + \frac{1}{x} \right )^{3} + \left ( x + \frac{1}{x} \right ) = 0$, is

  1. $\displaystyle 2$
  2. $\displaystyle 0$
  3. $\displaystyle 3$
  4. $\displaystyle 1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let y = x + 1/x. The equation is y^3 + y = 0, so y(y^2 + 1) = 0. Thus y = 0 or y^2 = -1. Since y must be real, y = 0. x + 1/x = 0 implies x^2 + 1 = 0, which has no real roots.

AI explanation

Factoring the equation by taking (x + 1/x) common gives (x + 1/x) * [(x + 1/x)^2 + 1] = 0. Because (x + 1/x)^2 is always positive for real x, the term [(x + 1/x)^2 + 1] is strictly greater than zero and cannot be zero. The remaining factor x + 1/x = 0 implies x^2 = -1, which yields no real solutions for x.

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