THE NATURE OF THE ROOTS OF A QUADRATIC EQUATION USING THE DISCRIMINANT || GRADE 9 MATHEMATICS Q1

Introduction

This tutorial will guide you through understanding the nature of the roots of a quadratic equation using the discriminant, a key concept in Grade 9 Mathematics. Knowing how to determine the nature of roots helps in solving equations and understanding their graphical representations.

Step 1: Understanding Quadratic Equations

• A quadratic equation is typically in the form of: [ ax^2 + bx + c = 0 ]
• Here, (a), (b), and (c) are constants, and (a \neq 0).
• The solutions (roots) of the quadratic equation can be found using the quadratic formula: [ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} ]

Step 2: Introducing the Discriminant

• The discriminant is the expression under the square root in the quadratic formula, given by: [ D = b^2 - 4ac ]
• The value of the discriminant determines the nature of the roots:
  • If (D > 0): Two distinct real roots
  • If (D = 0): One real root (a repeated root)
  • If (D < 0): No real roots (complex roots)

Step 3: Calculating the Discriminant

• To find the discriminant, follow these steps:
  1. Identify the coefficients (a), (b), and (c) from the quadratic equation.
  2. Substitute these values into the discriminant formula: [ D = b^2 - 4ac ]
  3. Calculate the value of (D).

Step 4: Analyzing the Roots

• Based on the value of the discriminant, analyze the roots:

  • Example with (D > 0):

    • For (2x^2 - 4x + 2 = 0):
      • Here, (a = 2), (b = -4), (c = 2)
      • Calculate (D = (-4)^2 - 4 \cdot 2 \cdot 2 = 16 - 16 = 0) (One real root)

  • Example with (D < 0):

    • For (x^2 + 4x + 8 = 0):
      • Here, (a = 1), (b = 4), (c = 8)
      • Calculate (D = 4^2 - 4 \cdot 1 \cdot 8 = 16 - 32 = -16) (No real roots)

Step 5: Graphical Interpretation

• Understanding the discriminant also helps in visualizing the quadratic equation:
  • A positive discriminant means the parabola intersects the x-axis at two points.
  • A zero discriminant means the parabola touches the x-axis at one point (vertex).
  • A negative discriminant means the parabola does not intersect the x-axis at all.

Conclusion

In this tutorial, you learned how to determine the nature of the roots of a quadratic equation using the discriminant. Remember to:

• Identify the coefficients of the quadratic equation.
• Calculate the discriminant and analyze its value.
• Use graphical interpretations to enhance your understanding of quadratic equations.

Practice with different quadratic equations to reinforce these concepts!