Lesson Note on Roots of Quadratic Equation: Meaning, Sum and Product of Roots for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 1st Term
Week: 1
Age: 16 years
Duration: 45 minutes
Subject: Further Mathematics
Curriculum Theme: Algebra and Equations
Previous Lesson: .
Topic: ROOTS OF QUADRATIC EQUATION
Subject Matter: Meaning of roots of a quadratic equation (values of x that satisfy ax^2 + bx + c = 0), sum and product of roots, forming quadratic equation given sum and product of roots, condition for quadratic equation to have equal roots (b^2 = 4ac), condition for quadratic equation to have real roots (b^2 > 4ac), condition for quadratic equation to have no real roots, complex roots (b^2 < 4ac)

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• Define the roots of a quadratic equation.
• State the formulas for the sum and product of roots of a quadratic equation.
• Explain the conditions for a quadratic equation to have equal, real, or complex roots.

Affective Domain:

• Appreciate the importance of understanding the nature of roots in solving mathematical problems.
• Develop interest in solving problems related to quadratic equations.

Psychomotor Domain:

• Calculate the sum and product of roots for given quadratic equations.
• Form quadratic equations when the sum and product of their roots are given.
• Determine the nature of roots for a quadratic equation using the discriminant.

Social Domain:

• Participate actively in classroom discussions and problem-solving activities.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• Further Mathematics for Senior Secondary Schools, Book 2

Instructional Materials

The teacher will teach this lesson with the aid of:

• Charts showing a quadratic equation and its general form.
• Whiteboard and markers.

Rationale for the Lesson

This lesson helps pupils understand the fundamental properties of quadratic equations, which is important for solving advanced mathematical problems. It enables pupils to analyze the solutions of quadratic equations without fully solving them, providing a deeper insight into algebraic structures.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of solving quadratic equations by factorization, completing the square, and using the general quadratic formula.

Lesson Content/Board Summary

ROOTS OF QUADRATIC EQUATION

Meaning of Roots of a Quadratic Equation

The roots of a quadratic equation are the values of the unknown variable (usually ‘x’) that satisfy the equation. For a general quadratic equation ax² + bx + c = 0 (where a ≠ 0), the roots are the values of x that make the equation true.

Sum and Product of Roots

For a quadratic equation in the form ax² + bx + c = 0, if α and β are its roots, then:

• Sum of roots (α + β) = -b/a
• Product of roots (αβ) = c/a

Where ‘a’ is the coefficient of x², ‘b’ is the coefficient of x, and ‘c’ is the constant term.

Worked Examples:

Example 1: Find the sum and product of the roots of the equation 2x² – 5x + 3 = 0.

Solution:

Step 1: Identify a, b, and c.

• a = 2
• b = -5
• c = 3

Step 2: Calculate the sum of roots.

• Sum = -b/a = -(-5)/2 = 5/2

Step 3: Calculate the product of roots.

• Product = c/a = 3/2

Example 2: Find the sum and product of the roots of the equation x² + 7x – 10 = 0.

Solution:

Step 1: Identify a, b, and c.

• a = 1
• b = 7
• c = -10

Step 2: Calculate the sum of roots.

• Sum = -b/a = -7/1 = -7

Step 3: Calculate the product of roots.

• Product = c/a = -10/1 = -10

Forming a Quadratic Equation from its Roots

If the sum (S) and product (P) of the roots of a quadratic equation are known, the equation can be formed using the formula:

• x² – (Sum of roots)x + (Product of roots) = 0
• x² – Sx + P = 0

Worked Examples:

Example 1: Form a quadratic equation whose roots have a sum of 4 and a product of -12.

Solution:

Step 1: Identify the given sum (S) and product (P).

• S = 4
• P = -12

Step 2: Substitute S and P into the formula x² – Sx + P = 0.

• x² – (4)x + (-12) = 0
• x² – 4x – 12 = 0

Example 2: Form a quadratic equation whose roots are 2 and -5.

Solution:

Step 1: Calculate the sum of the roots.

• Sum (S) = 2 + (-5) = -3

Step 2: Calculate the product of the roots.

• Product (P) = 2 × (-5) = -10

Step 3: Substitute S and P into the formula x² – Sx + P = 0.

• x² – (-3)x + (-10) = 0
• x² + 3x – 10 = 0

Nature of Roots (Using the Discriminant)

The nature of the roots of a quadratic equation ax² + bx + c = 0 can be determined by the value of the discriminant, denoted by Δ or D, where:

• Discriminant (Δ) = b² – 4ac

The conditions for the nature of the roots are:

• If b² – 4ac > 0: The roots are real and distinct (unequal).
• If b² – 4ac = 0: The roots are real and equal.
• If b² – 4ac < 0: The roots are complex (not real and unequal).

Worked Examples:

Example 1: Determine the nature of the roots of the equation 3x² – 7x + 2 = 0.

Solution:

Step 1: Identify a, b, and c.

• a = 3
• b = -7
• c = 2

Step 2: Calculate the discriminant (b² – 4ac).

• b² – 4ac = (-7)² – 4(3)(2)
• = 49 – 24
• = 25

Step 3: Interpret the result.

• Since 25 > 0, the roots are real and distinct.

Example 2: Determine the nature of the roots of the equation x² + 4x + 4 = 0.

Solution:

Step 1: Identify a, b, and c.

• a = 1
• b = 4
• c = 4

Step 2: Calculate the discriminant (b² – 4ac).

• b² – 4ac = (4)² – 4(1)(4)
• = 16 – 16
• = 0

Step 3: Interpret the result.

• Since 0 = 0, the roots are real and equal.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher revises previous knowledge on solving quadratic equations using the general formula and asks pupils to state the general form of a quadratic equation.
Pupils’ Activity: Pupils recall the general form ax² + bx + c = 0 and discuss methods of solving quadratic equations.
Learning Point: Pupils connect prior knowledge to the new topic.

Step 2: Meaning of Roots of a Quadratic Equation

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher explains the meaning of roots as the values of ‘x’ that satisfy the quadratic equation ax² + bx + c = 0.
Pupils’ Activity: Pupils listen attentively and ask questions for clarification.
Learning Point: Pupils understand what the roots of a quadratic equation represent.

Step 3: Sum and Product of Roots

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher leads pupils to derive and state the formulas for the sum (α + β = -b/a) and product (αβ = c/a) of the roots, explaining ‘a’, ‘b’, and ‘c’. The teacher works through examples on finding the sum and product of roots.
Pupils’ Activity: Pupils participate in the derivation, write down the formulas, and solve examples with the teacher.
Learning Point: Pupils learn how to calculate the sum and product of roots using coefficients.

Step 4: Forming a Quadratic Equation from its Roots

Time: 7 minutes
Teaching Skill: Demonstration
Teacher’s Activity: The teacher guides pupils to form quadratic equations given the sum and product of roots, using the formula x² – (Sum of roots)x + (Product of roots) = 0. The teacher provides worked examples.
Pupils’ Activity: Pupils follow the examples and practice forming equations.
Learning Point: Pupils learn to construct a quadratic equation from its roots.

Step 5: Nature of Roots (Introduction to Discriminant)

Time: 5 minutes
Teaching Skill: Explanation
Teacher’s Activity: The teacher introduces the discriminant, b² – 4ac, and explains how its value determines the nature of the roots (real and distinct, real and equal, or complex).
Pupils’ Activity: Pupils listen and note down the definition of the discriminant and its conditions.
Learning Point: Pupils understand the concept of the discriminant.

Step 6: Applying the Discriminant

Time: 8 minutes
Teaching Skill: Demonstration/Problem Solving
Teacher’s Activity: The teacher demonstrates how to apply the discriminant to determine the nature of roots for various quadratic equations, providing worked examples for each condition (b² – 4ac > 0, = 0, < 0).
Pupils’ Activity: Pupils observe the examples and attempt similar problems.
Learning Point: Pupils can determine the nature of roots using the discriminant.

Step 7: Evaluation/Review

Time: 5 minutes

Teaching Skill: Questioning/Assessment

Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define the roots of a quadratic equation.
2. For the equation 3x² + 8x – 5 = 0, state the sum and product of its roots.
3. Form a quadratic equation whose roots have a sum of -6 and a product of 9.
4. Determine the nature of the roots of the equation 2x² – 3x + 5 = 0.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 5 minutes
Teaching Skill: Summarizing
Teacher’s Activity: The teacher summarizes the key points of the lesson, reinforcing the formulas for sum and product of roots and the conditions for the nature of roots. The teacher gives an assignment to be submitted in the next class.
Pupils’ Activity: Pupils ask final questions and copy down the assignment.
Learning Point: Pupils consolidate their learning and prepare for further practice.

Lesson Keywords

• Roots – The values of the variable that satisfy a quadratic equation.
• Quadratic Equation – An equation of the form ax² + bx + c = 0, where a ≠ 0.
• Discriminant – The expression b² – 4ac, used to determine the nature of roots.
• Sum of Roots – The sum of the two solutions of a quadratic equation, given by -b/a.
• Product of Roots – The product of the two solutions of a quadratic equation, given by c/a.
• Real Roots – Roots that are real numbers.
• Complex Roots – Roots that are not real numbers, involving an imaginary part.
• Equal Roots – When both roots of a quadratic equation have the same value.

Differentiation

The teacher will provide additional examples and exercises for pupils who grasp concepts quickly, while offering one-on-one support and simplified problems for those needing more assistance. Group work will be used to encourage peer learning.

Note for teachers using this lesson plan

Teachers should ensure pupils are comfortable with basic algebraic manipulations before introducing the more abstract concepts of sum, product, and nature of roots. Emphasize the connection between the coefficients of the quadratic equation and the properties of its roots. Encourage pupils to practice with a variety of examples.