Nature of Roots

Understand the nature of roots of a quadratic equation using the discriminant, with clear explanations, cases, graphs, examples, and common mistakes.

1. Introduction

The nature of roots of a quadratic equation refers to the type of solutions the equation has: whether the roots are real, equal, distinct, or not real.

To determine the nature of roots without solving the entire equation, we use the discriminant, a key part of the quadratic formula.

2. The Discriminant

For a quadratic equation:

\( ax^2 + bx + c = 0 \)

The discriminant is defined as:

\( D = b^2 - 4ac \)

This value helps us quickly identify what kind of roots the equation has.

3. Cases Based on the Discriminant

The value of the discriminant decides the nature of the roots:

3.1. Case 1: D > 0 (Two Real and Distinct Roots)

If \(D > 0\), the term inside the square root is positive. Therefore, we get:

  • Two different real roots
  • Roots are irrational if D is not a perfect square

Example: \(x^2 - 5x + 6 = 0\) → \(D = 25 - 24 = 1\) → two distinct real roots.

3.2. Case 2: D = 0 (One Real and Equal Root)

If \(D = 0\), the square root term becomes 0. Thus we get:

  • Two equal real roots
  • The quadratic is a perfect square trinomial

Example: \(x^2 - 4x + 4 = 0\) → \(D = 16 - 16 = 0\) → root = 2 (repeated).

3.3. Case 3: D < 0 (No Real Roots)

If \(D < 0\), the value under the square root is negative. So:

  • No real roots are possible
  • Roots are complex (imaginary numbers)

Example: \(x^2 + x + 1 = 0\) → \(D = 1 - 4 = -3\).

4. Graphical Meaning of the Nature of Roots

The graph of a quadratic equation \(y = ax^2 + bx + c\) is a parabola. The discriminant tells us how the parabola interacts with the x-axis.

4.1. Graph for D > 0

The parabola cuts the x-axis at two points. These points correspond to the two real roots.

4.2. Graph for D = 0

The parabola touches the x-axis at exactly one point. This point is the repeated real root.

4.3. Graph for D < 0

The parabola does not touch the x-axis. Hence, there are no real roots.

5. Worked Examples

Let’s determine the nature of roots using the discriminant.

5.1. Example 1

Equation: \(x^2 - 2x - 8 = 0\)

\(D = (-2)^2 - 4(1)(-8) = 4 + 32 = 36\)

Since \(D > 0\), the equation has two distinct real roots.

5.2. Example 2

Equation: \(x^2 - 6x + 9 = 0\)

\(D = 36 - 36 = 0\)

Since \(D = 0\), the equation has one repeated real root.

5.3. Example 3

Equation: \(2x^2 + 3x + 5 = 0\)

\(D = 9 - 40 = -31\)

Since \(D < 0\), the equation has no real roots.

6. Importance of the Nature of Roots

  • Helps understand the behaviour of the quadratic equation without solving it.
  • Useful in graph analysis, factorisation, and determining real-world feasibility of solutions.
  • Essential for checking whether solutions exist in real numbers.

7. Common Mistakes

  • Incorrectly computing the discriminant due to sign errors.
  • Assuming D < 0 means one real root (actually no real root).
  • Forgetting to apply square root rule correctly.
  • Mixing up the meaning of D = 0 and D > 0.

8. Quick Practice

Find the nature of roots of the following equations:

  1. \(x^2 + 4x + 3 = 0\)
  2. \(x^2 - 2x + 1 = 0\)
  3. \(3x^2 + x + 2 = 0\)
  4. \(5x^2 - 7x + 2 = 0\)

9. Summary

  • The discriminant \(D = b^2 - 4ac\) determines the nature of roots.
  • If D > 0 → two real distinct roots.
  • If D = 0 → one repeated real root.
  • If D < 0 → no real roots.
  • Graphs visually show how the roots relate to x-axis intersections.