Lesson 1: Solving Quadratic Equations

👩🏽‍🏫 Introduction

Have you ever thrown a ball and watched it fly in the air? That curvy path it follows is a perfect example of a quadratic function at work!

Quadratic equations pop up in construction, sports, business, and even nature. In this lesson, we’ll learn how to solve quadratic equations — a key part of algebra that helps us predict outcomes and solve real problems.

By the end of this lesson, you’ll be able to:

• Identify a quadratic equation
• Solve quadratic equations using different methods
• Apply these methods to real-world problems

🧠 Key Concepts and Explanations

📌 What is a Quadratic Equation?

A quadratic equation is a second-degree equation in the form:

Where:

• a, b, and c are constants
• x is the unknown
• a ≠ 0

🧮 Methods for Solving Quadratic Equations

There are three main methods:

1. Factorisation
2. Completing the Square
3. Using the Quadratic Formula

🔢 Method 1: Factorisation

Try to express the equation as a product of two brackets and set each to zero.

Example:

1. Find two numbers that multiply to 6 and add to -5. They are -2 and -3.
2. So:
3. Set each factor to zero:

🧠 Method 2: Completing the Square

This method rewrites the equation in the form:

Example:

1. Move the constant:
2. Add to both sides:
   
3. Now:
4. Take square roots:
5. So:

📐 Method 3: Quadratic Formula

Use this when factorisation is hard. The formula is:

Example:

1. a = 2, b = 3, c = -2
2. Calculate discriminant:
3. Apply formula:
4. So:

✍🏽 Practice Exercises

🅰️ A. Solve by Factorisation

🅱️ B. Solve by Completing the Square

🆎 C. Solve Using the Quadratic Formula

✅ Answers and Explanations

🅰️ A. Factorisation

• 1.
• 2.

🅱️ B. Completing the Square

• 1.

🆎 C. Quadratic Formula

• 1. a = 3, b = -1, c = -2
  Discriminant = 1 + 24 = 25
  
• 2.

📌 Recap

• ✅ A quadratic equation is in the form
• ✅ We can solve quadratics by factorisation, completing the square, or the quadratic formula
• ✅ Choose the method based on simplicity, especially when factorising is not easy

💭 Reflection Prompt

If you were designing a parabolic bridge arch or planning a projectile’s path, which method would you use to solve the quadratic equation, and why?