Nature of Roots of a Quadratic equation

The discriminant of a quadratic equation determines the nature of the roots of the quadratic equation. The discriminant ($\Delta$ or $D$) is $b^2-4ac$ for the standard form quadratic equation $ax^2+bx+c = 0$.

There are six properties, which we need to understand to study the nature of roots of a quadratic equation. Learn all of them one after one with understandable examples.

Zero discriminant

The roots are real and equal if the discriminant of a quadratic equation is equal to zero.

Positive discriminant

The roots are two distinct real numbers if the discriminant is positive.

The roots are two distinct real and rational numbers if the discriminant is positive and can be expressed as a perfect square of another number, and the quadratic equation contains rational coefficients.

The roots are two distinct real and irrational numbers if the discriminant is positive and cannot be expressed as a perfect square of another number, and the quadratic equation contains rational coefficients.

Negative discriminant

The roots are different imaginary numbers and complex conjugates if the discriminant of a quadratic equation is less than zero.

There are two distinct roots if the discriminant is not equal to zero.