If the discriminant b^2 - 4ac is greater than zero, how many real solutions does the quadratic equation ax^2 + bx + c = 0 have?

• A. No real solutions.
• B. One real solution.
• C. Two complex conjugate solutions.
• D. Two real and distinct solutions.

Answer: (D)

Discriminant tells you how many real solutions a quadratic has. For ax^2 + bx + c = 0, the roots come from x = [-b ± sqrt(b^2 - 4ac)]/(2a). If the discriminant b^2 - 4ac is greater than zero, the square root term is a real number greater than zero, so the two roots produced by adding and subtracting this real square root are different. Since a ≠ 0 for a quadratic, dividing by 2a gives two distinct real solutions. If the discriminant were zero, there would be one real solution (a repeated root); if it were negative, the roots would be two complex conjugates. Therefore, a positive discriminant means two real and distinct solutions.