How many real roots does the quadratic equation x^2 + 2x + 5 = 0 have?

• A. 0
• B. 1
• C. 2
• D. Infinite

Answer: (A)

Think about how many real solutions a quadratic can have by looking at the discriminant, b^2 - 4ac. For x^2 + 2x + 5, a = 1, b = 2, c = 5. The discriminant is 2^2 - 4(1)(5) = 4 - 20 = -16, which is negative. A negative discriminant means the parabola never crosses the x-axis, so there are no real roots. You can also check by locating the vertex at x = -b/(2a) = -1; the value there is (-1)^2 + 2(-1) + 5 = 4, which is above zero, reinforcing that there are zero real zeros.