A quadratic equation can have which of the following numbers of real solutions?

• A. 2, 1, and 0
• B. 3
• C. 0
• D. 4

Answer: (A)

A quadratic equation can intersect the real number line at most twice. Whether you get two, one, or zero real solutions depends on the discriminant, b^2 - 4ac. If the discriminant is greater than zero, you have two distinct real solutions. If it equals zero, there is a single real solution (a repeated root). If it is less than zero, there are no real solutions because the parabola doesn’t cross the x-axis. So the possible numbers of real solutions are 2, 1, or 0. For example, x^2 - 5x + 6 = 0 has two real solutions (2 and 3); x^2 - 4x + 4 = 0 has one real solution (2); and x^2 + 1 = 0 has no real solutions. The options 3 or 4 real solutions can’t happen for a quadratic.