Which part of the quadratic formula determines the nature of the roots?

The correct choice is the part of the quadratic formula represented by the square root term, specifically √(b² - 4ac). This component is known as the discriminant, and it is key to determining the nature of the roots of the quadratic equation ax² + bx + c = 0.

The discriminant indicates whether the roots are real or complex, as well as whether they are distinct or repeated. If the value of the discriminant is greater than zero, it implies that there are two distinct real roots. If the discriminant equals zero, there is exactly one real root, which is a repeated (or double) root. Conversely, if the discriminant is less than zero, the roots are complex and not real.

In summary, this part of the formula is crucial because it directly affects how many real roots exist and their nature, helping to classify the solutions of the quadratic equation.