The equation x² – x – 2 = 0 in three dimensional space is represented by

The correct option is A pair of parallel planes.

Let us analyze the given equation step-by-step to understand its representation in three-dimensional space.

We are given the quadratic equation in one variable:
$$x^2-x-2=0$$

First, we can factorize this quadratic equation by splitting the middle term:
$$x^2-2x+x-2=0$$
$$x(x-2)+1(x-2)=0$$
$$(x-2)(x+1)=0$$

This gives two independent solutions for x:
$$x=2\text{\hspace{0.5em}or\hspace{0.5em}}x=-1$$

Now, let us interpret these equations in three-dimensional space (${ℝ}^3$), where any point is represented by coordinates $(x,y,z)$.

In three-dimensional space, the equation $x=c$ (where c is a constant) represents a plane parallel to the yz-plane. Since the variables y and z are absent from the equations, they can take any real value.

Therefore:
1. The equation $x=2$ represents a plane parallel to the yz-plane, passing through the point $(2,0,0)$.
2. The equation $x=-1$ represents another plane parallel to the yz-plane, passing through the point $(-1,0,0)$.

Since both planes are parallel to the same coordinate plane (the yz-plane), they are parallel to each other. Thus, the given quadratic equation represents a pair of parallel planes in three-dimensional space.