Given a quadratic equation with real coefficients, the discriminant (Δ) plays a crucial role in determining the nature of its roots. The discriminant of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
1. If [tex]\(\Delta > 0\)[/tex]:
- The roots are real and distinct.
- If [tex]\(\Delta\)[/tex] is a perfect square, the roots are rational.
- If [tex]\(\Delta\)[/tex] is not a perfect square, the roots are irrational.
2. If [tex]\(\Delta = 0\)[/tex]:
- The roots are real and equal.
3. If [tex]\(\Delta < 0\)[/tex]:
- The roots are complex (imaginary) and they occur as complex conjugates.
In this case, the given discriminant is [tex]\(-2\)[/tex]. Since the discriminant is less than zero ([tex]\(\Delta < 0\)[/tex]), the roots of the quadratic equation must be complex (imaginary). Therefore, the correct answer is:
2) imaginary
