Lesson 2: Forming Quadratic Equations from Roots

👨🏽‍🏫 Introduction

Imagine you’re a puzzle master 🔍 — and instead of solving the puzzle, you’re now building one! In this lesson, we’ll flip the usual approach: instead of solving quadratic equations, you’ll learn how to form them when you already know their solutions (also called roots).

It’s like reverse engineering — very useful when designing systems or checking if a pair of numbers could be the roots of an equation!

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

• Form a quadratic equation when given its roots
• Use the sum and product of roots to form the equation
• Apply the concept to real-life and theoretical problems

🧠 Key Concepts and Explanations

🔁 What Are Roots?

The roots of a quadratic equation are the values of x that satisfy the equation:

If you’re given the roots of the equation (say r₁ and r₂), you can form the equation by reversing the solving process.

🧩 Method 1: Using Factor Form

If r₁ and r₂ are the roots, the quadratic equation is:

Then expand the brackets and simplify.

🧠 Method 2: Using Sum and Product of Roots

For a quadratic equation in the form :

• Sum of roots:
• Product of roots:

So, if you know the sum and product of the roots, the equation is:

🧮 Sample Problem Walkthroughs

1. Form a quadratic equation with roots 3 and 5.
   Use:
   Expand:
2. Form a quadratic equation with roots -2 and 7.
   Use:
   Expand:
3. Form a quadratic equation where the sum of the roots is 6 and the product is 5.
   Use formula:
4. Form the equation with roots
   Use:
   Multiply brackets:
   To clear fractions, multiply through by 8:
   

✍🏽 Practice Exercises

🅰️ A. Using Root Pairs

1. Form a quadratic equation with roots 4 and -1.
2. Form a quadratic equation with roots -3 and -5.

🅱️ B. Using Sum and Product

1. Form a quadratic equation whose roots have a sum of 7 and a product of 10.
2. Form a quadratic equation whose roots have a sum of -4 and a product of -5.

🆎 C. With Fractions or Decimals

1. Form a quadratic equation with roots

✅ Answers and Explanations

🅰️ A. Using Root Pairs

• 1.
• 2.

🅱️ B. Using Sum and Product

• 1.
• 2.

🆎 C. Fractions

• 1. Roots:
  Equation:
  Expand:
  Multiply by 15:

📌 Recap

• ✅ To form a quadratic equation from roots, use and expand
• ✅ You can also use the sum and product of roots to form:
• ✅ Always clear fractions by multiplying through, if needed

💭 Reflection Prompt

Which method did you prefer — using root pairs or using the sum and product? Why might each be useful in different types of questions or careers?