To determine the nature of the roots of a quadratic equation, we rely on the value of its discriminant. The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex] is given by the formula:
[tex]\[
\Delta = b^2 - 4ac
\][/tex]
The value of the discriminant can tell us the nature of the roots of the quadratic equation:
1. If [tex]\(\Delta > 0\)[/tex], the quadratic equation has two distinct real roots.
2. If [tex]\(\Delta = 0\)[/tex], the quadratic equation has exactly one real root (or a repeated real root).
3. If [tex]\(\Delta < 0\)[/tex], the quadratic equation has two complex conjugate roots.
In this specific problem, we are given that the discriminant [tex]\(\Delta\)[/tex] of the quadratic equation is equal to [tex]\(-8\)[/tex]. Therefore, [tex]\(\Delta < 0\)[/tex], which implies that the quadratic equation has two complex roots.
Thus, the correct statement that describes the roots is:
- There are two complex roots.
