Consider the quadratic x^2 + 3x + 2 = 0. What is the nature of its roots?

Consider the quadratic x^2 + 3x + 2 = 0. What is the nature of its roots?

• A. One real root
• B. Two distinct real roots
• C. No real roots
• D. All real numbers

Answer: (B)

The question tests how the discriminant tells you how many real roots a quadratic has. For ax^2 + bx + c = 0, calculate Δ = b^2 − 4ac. If Δ > 0, you get two distinct real roots; if Δ = 0, a repeated real root; if Δ < 0, no real roots. Here, a = 1, b = 3, c = 2, so Δ = 3^2 − 4·1·2 = 9 − 8 = 1, which is positive. That means there are two real roots. Indeed, x^2 + 3x + 2 factors as (x + 1)(x + 2), giving roots x = −1 and x = −2, two distinct real numbers.

The question tests how the discriminant tells you how many real roots a quadratic has. For ax^2 + bx + c = 0, calculate Δ = b^2 − 4ac. If Δ > 0, you get two distinct real roots; if Δ = 0, a repeated real root; if Δ < 0, no real roots.

Here, a = 1, b = 3, c = 2, so Δ = 3^2 − 4·1·2 = 9 − 8 = 1, which is positive. That means there are two real roots. Indeed, x^2 + 3x + 2 factors as (x + 1)(x + 2), giving roots x = −1 and x = −2, two distinct real numbers.