Lesson Note on Quadratic Equation: Completing the Square and Formula for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: 1st Term
Week: 8
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Algebra
Previous Lesson: Geometrical Progression: Nth Term, Mean and Sum.
Topic: QUADRATIC EQUATION (I)
Subject Matter: Revision of factorization, finding what should be added to an algebraic expression to make it a perfect square, quadratic equation using completing the square method, deducing the quadratic formula from completing the square and its application to solving problems.

Specific Objectives

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

Cognitive Domain:

• Recall the process of factorizing perfect square trinomials.
• Identify the constant term needed to make a quadratic expression a perfect square.
• State the steps for solving quadratic equations using the completing the square method.
• Derive the quadratic formula from the completing the square method.
• Solve quadratic equations using the quadratic formula.

Affective Domain:

• Appreciate the importance of factorization in solving quadratic equations.
• Develop patience and precision when working through algebraic derivations.
• Show interest in applying different methods to solve quadratic equations.

Psychomotor Domain:

• Factorize given perfect square expressions accurately.
• Construct perfect square trinomials by adding the correct constant.
• Apply the completing the square method to solve quadratic equations.
• Demonstrate the derivation of the quadratic formula on the board.
• Accurately use the quadratic formula to find solutions to quadratic equations.

Social Domain:

• Collaborate with peers to solve quadratic problems.
• Participate actively in class discussions on algebraic techniques.
• Present solutions clearly and logically to the class.

Reference Materials

The following resources were used in planning this lesson:

• Senior Secondary Schools Education Curriculum
• State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 2
• https://www.mathsisfun.com/algebra/quadratic-equation-completing-square.html
• https://www.youtube.com/watch?v=bNj5rM6wY48

Instructional Materials

The teacher will teach this lesson with the aid of:

• Quadratic equation box
• Completing the squares sheet
• Whiteboard and markers

Rationale for the Lesson

This lesson helps pupils understand different methods for solving quadratic equations, which are fundamental in higher mathematics and various scientific fields. It builds a strong foundation for future topics and enables pupils to choose the most efficient method for solving problems.

Prerequisite/Previous Knowledge

Pupils have prior knowledge of basic algebraic operations, factorization of simple quadratic expressions, and expanding algebraic expressions.

Lesson Content/Board Summary

QUADRATIC EQUATION (I)

Definition of a Quadratic Equation

A quadratic equation is an equation of the form ax² + bx + c = 0, where ‘a’, ‘b’, and ‘c’ are constants, and ‘a’ is not equal to zero (a ≠ 0). The highest power of the variable (x) is 2.

Revision of Factorization of Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that results from squaring a binomial. It has the form a² ± 2ab + b², which factorizes to (a ± b)².

Examples:

1. x² + 6x + 9 = (x + 3)²
2. y² – 10y + 25 = (y – 5)²
3. 4x² + 12x + 9 = (2x + 3)²

Finding the Constant to Make a Quadratic Expression a Perfect Square

For an expression in the form x² + bx + k to be a perfect square, the constant term ‘k’ must be equal to (b/2)².

Worked Example 1: Find the value of ‘k’ that makes x² + 8x + k a perfect square.

Solution:
Step 1: Identify ‘b’. Here, b = 8.
Step 2: Calculate k using the formula k = (b/2)².
k = (8/2)² = 4² = 16.
Therefore, x² + 8x + 16 is a perfect square, which factorizes to (x + 4)².

Worked Example 2: Find the value of ‘k’ that makes x² – 12x + k a perfect square.

Solution:
Step 1: Identify ‘b’. Here, b = -12.
Step 2: Calculate k using the formula k = (b/2)².
k = (-12/2)² = (-6)² = 36.
Therefore, x² – 12x + 36 is a perfect square, which factorizes to (x – 6)².

Solving Quadratic Equations by Completing the Square Method

This method converts a quadratic equation into a perfect square trinomial on one side, allowing for easy solution by taking the square root.

Steps:

1. Ensure the coefficient of x² is 1. If not, divide the entire equation by the coefficient of x².
2. Move the constant term to the right-hand side of the equation.
3. Take half of the coefficient of x, square it, and add it to both sides of the equation.
4. Factorize the left-hand side as a perfect square.
5. Take the square root of both sides (remembering ±).
6. Solve for x.

Worked Example 1: Solve x² + 6x + 5 = 0 by completing the square.

Solution:
Step 1: Coefficient of x² is 1. (Already done)
Step 2: Move the constant term to the RHS.
x² + 6x = -5
Step 3: Half of coefficient of x is (6/2) = 3. Square it: 3² = 9. Add 9 to both sides.
x² + 6x + 9 = -5 + 9
Step 4: Factorize the LHS.
(x + 3)² = 4
Step 5: Take the square root of both sides.
√(x + 3)² = ±√4
x + 3 = ±2
Step 6: Solve for x.
x + 3 = 2 or x + 3 = -2
x = 2 – 3 or x = -2 – 3
x = -1 or x = -5
The solutions are x = -1 or x = -5.

Worked Example 2: Solve 2x² – 5x – 3 = 0 by completing the square.

Solution:
Step 1: Divide by the coefficient of x² (which is 2).
(2x² – 5x – 3) / 2 = 0 / 2
x² – (5/2)x – 3/2 = 0
Step 2: Move the constant term to the RHS.
x² – (5/2)x = 3/2
Step 3: Half of coefficient of x is (-5/2)/2 = -5/4. Square it: (-5/4)² = 25/16. Add 25/16 to both sides.
x² – (5/2)x + 25/16 = 3/2 + 25/16
x² – (5/2)x + 25/16 = 24/16 + 25/16
x² – (5/2)x + 25/16 = 49/16
Step 4: Factorize the LHS.
(x – 5/4)² = 49/16
Step 5: Take the square root of both sides.
√(x – 5/4)² = ±√(49/16)
x – 5/4 = ±7/4
Step 6: Solve for x.
x – 5/4 = 7/4 or x – 5/4 = -7/4
x = 7/4 + 5/4 or x = -7/4 + 5/4
x = 12/4 or x = -2/4
x = 3 or x = -1/2
The solutions are x = 3 or x = -1/2.

Deriving the Quadratic Formula from Completing the Square

The quadratic formula is a general solution for any quadratic equation ax² + bx + c = 0. It can be derived using the completing the square method.

Derivation:
Start with the general quadratic equation: ax² + bx + c = 0

1. Divide by ‘a’ (assuming a ≠ 0):
   x² + (b/a)x + (c/a) = 0
2. Move the constant term to the RHS:
   x² + (b/a)x = -c/a
3. Add (half of coefficient of x)² to both sides. Half of (b/a) is (b/2a). Square it: (b/2a)².
   x² + (b/a)x + (b/2a)² = -c/a + (b/2a)²
   x² + (b/a)x + b²/4a² = -c/a + b²/4a²
4. Factorize the LHS as a perfect square and simplify the RHS:
   (x + b/2a)² = (b² – 4ac) / 4a²
5. Take the square root of both sides:
   x + b/2a = ±√((b² – 4ac) / 4a²)
   x + b/2a = ±√(b² – 4ac) / √(4a²)
   x + b/2a = ±√(b² – 4ac) / 2a
6. Solve for x:
   x = -b/2a ± √(b² – 4ac) / 2a
   Combine the terms:
   x = (-b ± √(b² – 4ac)) / 2a

This is the quadratic formula.

Application of the Quadratic Formula

The quadratic formula can be used to solve any quadratic equation directly.

Formula: x = (-b ± √(b² – 4ac)) / 2a

Worked Example 1: Solve x² – 5x + 6 = 0 using the quadratic formula.

Solution:
Step 1: Identify a, b, c. Here, a = 1, b = -5, c = 6.
Step 2: Substitute values into the formula.
x = (-(-5) ± √((-5)² – 4 * 1 * 6)) / (2 * 1)
x = (5 ± √(25 – 24)) / 2
x = (5 ± √1) / 2
x = (5 ± 1) / 2
Step 3: Calculate the two possible values for x.
x1 = (5 + 1) / 2 = 6 / 2 = 3
x2 = (5 – 1) / 2 = 4 / 2 = 2
The solutions are x = 3 or x = 2.

Worked Example 2: Solve 3x² + 7x + 2 = 0 using the quadratic formula.

Solution:
Step 1: Identify a, b, c. Here, a = 3, b = 7, c = 2.
Step 2: Substitute values into the formula.
x = (-7 ± √(7² – 4 * 3 * 2)) / (2 * 3)
x = (-7 ± √(49 – 24)) / 6
x = (-7 ± √25) / 6
x = (-7 ± 5) / 6
Step 3: Calculate the two possible values for x.
x1 = (-7 + 5) / 6 = -2 / 6 = -1/3
x2 = (-7 – 5) / 6 = -12 / 6 = -2
The solutions are x = -1/3 or x = -2.

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 greets the pupils and reviews the previous lesson on factorization of simple quadratic expressions. The teacher then introduces the topic of quadratic equations, stating its importance.
Pupils’ Activity: Pupils respond to greetings and questions, recalling previous knowledge. They listen attentively to the introduction.
Learning Point: Pupils connect previous knowledge to the new topic and understand the relevance of quadratic equations.

Step 2: Revision of Factorization of Perfect Squares

Time: 7 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher revises how to factorize perfect square trinomials, providing examples like x² + 10x + 25 and 4y² – 12y + 9. The teacher leads pupils to realize that all perfect squares are factorizable.
Pupils’ Activity: Pupils expand and factorize perfect squares, providing examples and confirming that perfect squares can always be factorized.
Learning Point: Pupils reinforce their understanding of perfect square factorization.

Step 3: Making an Expression a Perfect Square

Time: 7 minutes
Teaching Skill: Guided Practice
Teacher’s Activity: The teacher guides pupils on how to find the constant ‘k’ that makes a quadratic expression (x² + bx + k) a perfect square, using the formula k = (b/2)². The teacher provides examples for pupils to work through.
Pupils’ Activity: Pupils follow the teacher’s examples and practice finding the constant ‘k’ for various expressions.
Learning Point: Pupils learn to identify and add the correct constant term to complete the square.

Step 4: Solving Quadratic Equations by Completing the Square Method

Time: 8 minutes
Teaching Skill: Demonstration/Explanation
Teacher’s Activity: The teacher guides pupils through the steps involved in solving quadratic equations using the completing the square method. The teacher solves two worked examples on the board, explaining each step clearly.
Pupils’ Activity: Pupils pay close attention to the steps and examples. They participate by suggesting steps and asking clarifying questions.
Learning Point: Pupils understand and apply the completing the square method to solve quadratic equations.

Step 5: Deducing the Quadratic Formula

Time: 7 minutes
Teaching Skill: Derivation/Lecture
Teacher’s Activity: The teacher leads pupils through the step-by-step derivation of the quadratic formula from the general quadratic equation (ax² + bx + c = 0) using the completing the square method.
Pupils’ Activity: Pupils follow the derivation process, copying the steps into their notes and contributing to each step.
Learning Point: Pupils understand the origin of the quadratic formula and how it is derived.

Step 6: Application of the Quadratic Formula

Time: 5 minutes
Teaching Skill: Problem Solving
Teacher’s Activity: The teacher demonstrates how to apply the derived quadratic formula to solve quadratic equations by working through two examples on the board.
Pupils’ Activity: Pupils observe the application of the formula and attempt to solve similar problems.
Learning Point: Pupils learn to use the quadratic formula as an alternative method for solving quadratic equations.

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 a quadratic equation.
2. What constant term must be added to x² – 14x to make it a perfect square?
3. State the steps for solving a quadratic equation by completing the square.
4. Derive the quadratic formula from ax² + bx + c = 0.
5. Solve the equation x² + 2x – 8 = 0 using the quadratic formula.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 8: Conclusion

Time: 1 minute
Teaching Skill: Summarization
Teacher’s Activity: The teacher summarizes the key learning points of the lesson, emphasizing the different methods for solving quadratic equations (factorization, completing the square, and quadratic formula). The teacher gives pupils an assignment to practice these methods.
Pupils’ Activity: Pupils listen to the summary and copy down the assignment.
Learning Point: Pupils consolidate their understanding and are encouraged to practice.

Lesson Keywords

• Quadratic Equation – An equation of the form ax² + bx + c = 0, where a ≠ 0.
• Factorization – The process of breaking down an expression into a product of its factors.
• Perfect Square Trinomial – A trinomial that is the square of a binomial, e.g., (x + a)².
• Completing the Square – A method used to solve quadratic equations by transforming one side of the equation into a perfect square trinomial.
• Quadratic Formula – A general formula used to solve any quadratic equation: x = (-b ± √(b² – 4ac)) / 2a.

Differentiation

For struggling learners, the teacher will provide additional guided practice with simpler examples and step-by-step prompts. Advanced learners will be given more complex quadratic equations to solve and encouraged to explore applications in real-world problems.

Note for teachers using this lesson plan

Ensure pupils have a solid grasp of basic algebraic manipulations before introducing these methods. Emphasize the connection between completing the square and the derivation of the quadratic formula. Encourage pupils to practice both methods to gain proficiency.