How do I decide the nature of roots using the discriminant b² − 4ac?

The expression b² − 4ac is called the discriminant, usually written as D. For a quadratic ax² + bx + c = 0, just calculate D and check its sign. If D is greater than 0, the equation has two distinct real roots. If D equals exactly 0, the roots are real and equal (one repeated root). If D is less than 0, there are no real roots, the roots are imaginary. For example, in x² − 4x + 4 = 0, D = 16 − 16 = 0, so equal roots. In x² + x + 1 = 0, D = 1 − 4 = −3, less than 0, so no real roots. Always compute D first before deciding the nature.