Nature of Roots of a Quadratic Equation

The nature of the roots of a quadratic equation describes the characteristics of its solutions (or roots), indicating if they are real and distinct, real and equal, or purely imaginary. By using the discriminant, you can easily classify the roots and solve equations effectively. In this guide, we will learn the formula to find the nature of roots of quadratic equation, and solve examples to build a strong understanding of the concept.

Table of Contents

• What are the Natures of Roots of a Quadratic Equation?
• Solved Examples on Nature of Roots of a Quadratic Equation
• Practice Questions on Nature of Roots of a Quadratic Equation

What are the Natures of Roots of a Quadratic Equation

The nature of the roots of a quadratic equation tells us the type of solutions an equation will yield without actually solving it. A quadratic equation ax² + bx + c = 0 will have two roots.

For the quadratic equation ax² + bx + c = 0, the discriminant D = b² - 4ac determines the nature of the roots.

• If D > 0: √D is a positive real number, ∴ two distinct real roots. The roots are x = (−b + √D)/2a and x = (−b − √D)/2a

• If D = 0: √D = 0, ∴ both roots equal, and x = −b/(2a)

• If D < 0: √D is not a real number, ∴ no real roots exist, and all the roots are imaginary. The roots are complex conjugates.

• If D is a perfect square, then roots are rational. 

• If D is not a perfect square, then the roots are irrational (conjugate surds: p ± √q)
  
  

Solved Examples on Nature of Roots of a Quadratic Equation

Example 1: Determine the nature of the roots of the quadratic equation x²−4x+4=0.
Solution: a = 1, b = -4 and c = 4.
Discriminant, D =  b² - 4ac  = (-4)² - 4 × 1 × 4 = 16 - 16 = 0
Since D=0, the roots are real and equal.

Example 2: Determine the nature of the roots of the quadratic equation x²+5x+6=0
Solution: a = 1, b = 5 and c = 6
Discriminant, D =  b² - 4ac  = (5)² - 4 × 1 × 5 = 25 - 20 = 5
Since D > 0, the roots are distinct real roots.

Example 3: Determine the nature of the roots of the quadratic equation: 3x²−2x+1=0
Solution:  a = 3, b = -2 and c = 1
Discriminant, D =  b² - 4ac  = (-2)² - 4 × 3 × 1 = 4 - 12 = -8
Since D < 0, the roots are imaginary roots and conjugate.

Practice Questions on Nature of Roots of a Quadratic Equation

1. Find the nature of roots of 2x² + 2x − 3 = 0.

2. Find the nature of roots of x² − 2x − 7 = 0.

3. Find the nature of roots of 3x² + 6x + 2 = 0.