Quadratic Equations
Solving Quadratic Equations Using Different Methods
Learning Standards
Content Standard
The learner demonstrates understanding of key concepts of quadratic equations, inequalities and functions
Performance Standard
The learner is able to investigate mathematical relationships in various situations involving quadratic equations and solve these using a variety of strategies
Learning Competency
Solve quadratic equations by factoring, completing the square, and using the quadratic formula
Code: M9AL-Ia-b-1
Complete Lesson Plan
Learning Objectives
- •
Define quadratic equations in standard form
- •
Solve quadratic equations by factoring
- •
Solve quadratic equations by completing the square
- •
Apply the quadratic formula to solve equations
Lesson Procedures
motivation
Puzzle: 'A number multiplied by itself plus 5 times that number equals 36. What is the number?' Show that this creates: x² + 5x = 36, a quadratic equation.
presentation
- •
METHOD 1: SOLVING BY FACTORING
Steps:
- Write in standard form: ax² + bx + c = 0
- Factor the quadratic expression
- Apply Zero Product Property: If AB = 0, then A = 0 or B = 0
- Solve for x
Example 1: x² + 5x + 6 = 0
- Factor: (x + 2)(x + 3) = 0
- Set each factor = 0:
- x + 2 = 0 → x = -2
- x + 3 = 0 → x = -3
- Solutions: x = -2 or x = -3
Example 2: x² - 4 = 0
- Factor: (x + 2)(x - 2) = 0
- Solutions: x = 2 or x = -2
- •
METHOD 2: COMPLETING THE SQUARE
Steps:
- Move constant to right side
- Divide by coefficient of x² (if not 1)
- Add (b/2)² to both sides
- Factor left side as perfect square
- Take square root of both sides
- Solve for x
Example: x² + 6x + 5 = 0
- x² + 6x = -5
- Add (6/2)² = 9 to both sides:
- x² + 6x + 9 = -5 + 9
- x² + 6x + 9 = 4
- Factor: (x + 3)² = 4
- Take square root: x + 3 = ±2
- Solutions:
- x + 3 = 2 → x = -1
- x + 3 = -2 → x = -5
- •
METHOD 3: QUADRATIC FORMULA
For ax² + bx + c = 0:
x = [-b ± √(b² - 4ac)] / 2a
Steps:
- Identify a, b, c
- Substitute into formula
- Simplify
Example: 2x² + 3x - 2 = 0
- a = 2, b = 3, c = -2
- x = [-3 ± √(3² - 4(2)(-2))] / 2(2)
- x = [-3 ± √(9 + 16)] / 4
- x = [-3 ± √25] / 4
- x = [-3 ± 5] / 4
Solutions:
- x = (-3 + 5)/4 = 2/4 = 1/2
- x = (-3 - 5)/4 = -8/4 = -2
- •
THE DISCRIMINANT (b² - 4ac)
- If b² - 4ac > 0: Two real solutions
- If b² - 4ac = 0: One real solution (repeated root)
- If b² - 4ac < 0: No real solutions (imaginary)
generalization
Questions:
- What is a quadratic equation?
- What are the three methods for solving quadratic equations?
- When is factoring the easiest method?
- When should we use the quadratic formula?
- What does the discriminant tell us?
guided practice
- •
Solve together using all three methods:
Equation: x² - 3x - 10 = 0
By Factoring:
- (x - 5)(x + 2) = 0
- x = 5 or x = -2
- •
By Completing the Square:
- x² - 3x = 10
- x² - 3x + (3/2)² = 10 + 9/4
- (x - 3/2)² = 49/4
- x - 3/2 = ±7/2
- x = 5 or x = -2
- •
By Quadratic Formula:
- a = 1, b = -3, c = -10
- x = [3 ± √(9 + 40)] / 2
- x = [3 ± 7] / 2
- x = 5 or x = -2
- •
Practice: 2x² + 7x + 3 = 0
independent practice
- •
Activity 1: Solve by factoring:
- x² - 9 = 0
- x² + 7x + 12 = 0
- x² - 5x + 6 = 0
- •
Activity 2: Solve by completing the square:
- x² + 4x + 3 = 0
- x² - 6x + 8 = 0
- •
Activity 3: Solve using the quadratic formula:
- 3x² + 2x - 1 = 0
- x² - 4x + 1 = 0
- 2x² + 5x - 3 = 0
- •
Activity 4: Find the discriminant and determine the nature of roots:
- x² + 6x + 9 = 0
- x² + x + 1 = 0
- 2x² - 3x - 5 = 0
preliminary activities
- •
Prayer and greetings
- •
Review: Factoring polynomials, perfect square trinomials
- •
Quick drill: Factor: x² + 5x + 6
Assessment
answers
- x = -3 or x = -5
- x = 7 or x = 3
- x = 1 or x = 2/3
- Discriminant = -16 (No real solutions)
- w² + 2w - 24 = 0; Solutions: w = 4 (width = 4cm, length = 6cm)
evaluation
- •
Solve using any method:
- x² + 8x + 15 = 0 (by factoring)
- x² - 10x + 21 = 0 (by completing the square)
- 3x² - 5x + 2 = 0 (using quadratic formula)
- Find the discriminant of x² + 2x + 5 = 0 and describe the roots
- Word Problem: The area of a rectangle is 24 cm². The length is 2 cm more than the width. Find the dimensions. (Set up and solve)
Materials & Resources
- •
Algebra tiles (optional)
- •
Scientific calculator
- •
Quadratic formula reference cards
- •
Manila paper and markers
- •
Practice worksheets
- •
Step-by-step solution guides
assignment
Homework:
- Solve 5 quadratic equations using each method
- Complete textbook exercises on page ___
- Research: Real-life applications of quadratic equations
Remarks:
- Emphasize checking solutions by substitution
- Provide formula card for quadratic formula
subject matter
Topic: Solving Quadratic Equations
Key Concepts:
- Quadratic equation: ax² + bx + c = 0 (where a ≠ 0)
- Standard form: ax² + bx + c = 0
- Solutions/Roots: Values of x that satisfy the equation
- Three methods:
- Factoring (when equation can be factored)
- Completing the square (works for all equations)
- Quadratic formula (works for all equations)
Materials:
- Algebra tiles (if available)
- Calculator
- Formula cards
- Manila paper and markers
- Practice worksheets