In the real number system, how many solutions can a quadratic equation have?

• A. Exactly two real solutions
• B. Infinitely many real solutions
• C. Exactly one real solution
• D. Zero, one, or two real solutions

Answer: (D)

Verified step by step guidance

1. Recall that a quadratic equation in the real number system is generally written as \(ax^2 + bx + c = 0\), where \(a \neq 0\).
2. The number of real solutions depends on the discriminant, which is calculated as \(\Delta = b^2 - 4ac\).
3. If \(\Delta > 0\), the quadratic equation has exactly two distinct real solutions.
4. If \(\Delta = 0\), the quadratic equation has exactly one real solution (a repeated root).
5. If \(\Delta < 0\), the quadratic equation has no real solutions (the solutions are complex or imaginary). Therefore, a quadratic equation can have zero, one, or two real solutions.