To determine the number of real solutions of a quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex], we use the discriminant, which is calculated as [tex]\( \Delta = b^2 - 4ac \)[/tex]. The discriminant helps us understand the nature of the roots of the quadratic equation:
1. If [tex]\( \Delta > 0 \)[/tex], the equation has two distinct real solutions.
2. If [tex]\( \Delta = 0 \)[/tex], the equation has exactly one real solution (a repeated root).
3. If [tex]\( \Delta < 0 \)[/tex], the equation has no real solutions (the solutions are complex or imaginary).
In this particular problem, the discriminant value is given as [tex]\(\Delta = -16\)[/tex].
Since [tex]\(\Delta = -16\)[/tex] is less than zero ([tex]\(\Delta < 0\)[/tex]), the quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] has no real number solutions. Specifically, the roots of the equation are complex numbers because the discriminant is negative.
Therefore, the number of real number solutions for the quadratic equation is [tex]\(\boxed{0}\)[/tex].
