Different Cases of Nature of Roots
The nature of roots depends on the value of the Discriminant obtained for a given quadratic equation. Hence, the different cases of the nature of roots has been listed below:
- D > 0
- D = 0
- D < 0
- D is Perfect Square
- D is not Perfect Square
These conditions for nature of roots have been discussed extensively in the article below:
D > 0 (Positive Discriminant)
- Two distinct real roots mean the quadratic equation has two different real solutions.
- Here the discrimination will be positive.
D = 0 (Zero Discriminant)
- One real root: In this case, the quadratic equation has only one real solution, and this solution is repeated.
- Here the discriminant will be equal to zero.
D < 0 (Negative Discriminant)
- No real roots: The quadratic equation has no real solutions. Instead, it has two complex (conjugate) roots, which are of the form “a + bi” and “a – bi,” where “a” and “b” are real numbers, and “i” is the imaginary unit.
- Here the discriminant will be negative.
D is a Perfect Square
- When the discriminant (D) of a quadratic equation is a perfect square (the square of a rational number), the equation has rational (real) roots.
- Example: If D = 25, which is 52, it’s a perfect square discriminant. The equation has real roots: x = (-b ± 5) / (2a).
D is not a Perfect Square
- When D is not a perfect square, it leads to quadratic equations with either distinct irrational roots or complex conjugate roots.
- Example: For D = 8, which is not a perfect square, the equation has two distinct irrational roots: x = (-b ± √8) / (2a)
Nature of Roots
Roots are the solutions of an equation. The Nature of Roots in mathematics refers to the characteristics and properties of solutions to algebraic equations. These roots represent the values that make the equation true. Understanding the nature of roots is essential for solving equations in science and engineering to analyzing data in statistics. Depending on the equation, roots can be real or complex, and their behavior can provide insights into mathematical relationships. Our context of root in this article is for Quadratic Equations. Nature of Roots is important for Class 10 students.
In this article, we will learn about what are the roots of a quadratic equation, how to determine the nature of roots of a quadratic equation specifying different cases, and solve examples based on the nature of roots.
Table of Content
- What are the Roots of Quadratic Equation?
- Nature of Roots of Quadratic Equation
- Different Cases of Nature of Roots
- Nature of Roots – Summary
- Solved Examples