Negative discriminant: no real roots

Determine whether the quadratic equation x² + x + 4 = 0 has real roots, and explain your reasoning.

Step-by-step

1. a = 1, b = 1, c = 4
   First read off the three coefficients, paying attention to the sign of the constant term.
2. Δ = b² − 4ac = 1² − 4 × 1 × 4 = -15
   The discriminant determines the number of real roots: greater than zero gives two distinct real roots, equal to zero gives two equal real roots, less than zero gives no real roots.
3. Δ = -15 is less than zero ⇒ no real roots
   A negative discriminant means the graph does not meet the x-axis; Vieta's relation x_1 + x_2 = −b/a = -1 still holds over the complex numbers

Why it works

A negative discriminant indicates that the parabola lies entirely above or below the x-axis, has no intersection with the x-axis, and therefore has no real roots. At this point, do not blindly apply the quadratic formula: taking the square root of a negative number is meaningless in the real number system. Note that having no real roots does not mean the equation is meaningless; it simply means its roots lie in the complex number range.

Common mistakes

• Blindly applying the quadratic formula despite a negative discriminant, and taking the square root of a negative number.
• Saying "the equation has no solution" instead of "no real roots"; these two phrases have different meanings.
• Calculating the discriminant correctly but forgetting to answer the question's specific query about whether there are real roots.

Related

n-math-quadratic-discriminant-param n-math-quadratic-roots-1 n-math-quadratic-roots-2