Lesson 3: Word Problems Involving Quadratics

👨🏽‍🏫 Introduction

Have you ever seen a rectangular garden, a speeding car, or a number puzzle in an exam question? 🎯 These everyday situations can often be turned into quadratic equations — and solving them helps us make sense of real-life scenarios.

In this lesson, we’ll practise building quadratic equations from word problems, and then solving them step by step.

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

• Translate real-life problems into quadratic equations
• Use symbols and algebra to model word problems
• Solve and interpret your answers in context

🧠 Key Concepts and Explanations

📌 Steps for Solving Word Problems

1. Read carefully – Understand what’s given and what is unknown.
2. Define variables – Use x to represent the unknown quantity.
3. Write an equation – Translate words into a quadratic equation.
4. Solve the equation – Use factorisation, completing the square, or formula.
5. Check your answers – Make sure they make sense in the context.

🧮 Sample Problem Walkthroughs

1. The product of two consecutive integers is 132. What are the integers?
   Let the first number be x, so the next is x + 1.
   Equation:
   
   Expand:
   
   Factor:
   
   So, x = -12 or x = 11
   Therefore, the two integers are either -12 & -11 or 11 & 12
2. The length of a rectangle is 3 m more than its width. If the area is 40 m², find its dimensions.
   Let width = x m. Then length = x + 3 m
   Area =
   
   Expand:
   
   Factor:
   
   So, x = -8 (not valid) or x = 5
   Width = 5 m, Length = 8 m ✅
3. A car travels 60 km at a certain speed. If the speed were 20 km/h faster, the trip would take 1 hour less. Find the original speed.
   Let the speed be x km/h
   Then time =
   
   and new time =
   
   So,
   
   Multiply through by :
   
   Simplify:
   
   Rearranged:
   
   Solve:
   
   So, the original speed = 20 km/h ✅

✍🏽 Practice Exercises

🅰️ A. Basic Translation

1. The square of a number is 64 more than the number. Find the number.
2. The product of two numbers is 120 and one is 4 more than the other. Find the numbers.

🅱️ B. Geometry and Motion

1. The length of a rectangle is twice its width. If the area is 72 cm², find the dimensions.
2. A train travels 240 km at a certain speed. If the speed had been 20 km/h faster, the journey would have taken 1 hour less. Find the original speed.

🆎 C. Number Puzzles

1. The sum of two numbers is 13 and their product is 40. Find the numbers.

✅ Answers and Explanations

🅰️ A. Basic Translation

• 1. Let number be x:
  
  Solving:
• 2. Let number be x, then other = x + 4:
  
  Solving:

🅱️ B. Geometry and Motion

• 1. Let width = x, length = 2x:
  
  Dimensions = 6 cm × 12 cm
• 2. Let speed = x:
  
  Multiply through and simplify:
  
  Speed = 60 km/h

🆎 C. Number Puzzles

• Let numbers be x and 13 – x:
  
  Solving:

📌 Recap

• ✅ Word problems can be modelled using quadratic equations
• ✅ Always define variables clearly and form the equation step by step
• ✅ Solve using appropriate method and interpret in context

💭 Reflection Prompt

Which type of word problem did you find most interesting — numbers, shapes, or travel? How might this skill be useful outside the classroom?