Quadratic equations have been a cornerstone of algebra for centuries, from projectile motion in physics to revenue prediction in economics. A quadratic equation is any equation that can be written in the standard form ax²+bx+c=0, where a, b, and c are real numbers and a≠0. The term “quadratic” comes from the Latin word quadratus, meaning square, because the variable is squared. The systematic study of quadratics began in ancient Babylon. Still, the general quadratic formula we use today was first introduced by the Persian mathematician Muhammad ibn Musa al-Khwarizmi in the 9th century, who is also considered the “father of algebra.” He pioneered techniques to solve quadratic equations that laid the foundation for modern algebraic thinking. Over time, these equations have found wide applications in real-world scenarios, known as quadratic equations word problems, and are a vital part of the IGCSE Mathematics curriculum.
Let’s dive into this fascinating topic with progressively challenging solved examples to build your concept clarity.
Examples of Quadratic Equations Word Problems
🧠 Example 1: Basic Geometry Application
Problem:
The area of a rectangular garden is 77 m². Its length is 11 meters more than its width. Find the dimensions of the garden.
Solution:
Let width = ‘x’ meters
Then length = ‘x+11’ meters
Area = length × width = x(x+11)=77 → x²+11x−77=0
Using Mid-term splitting factorization: (You can use the quadratic formula or completing the square too)
(x+11)(x−7)=0
So, x=−11 or x=7
Reject negative width.
Answer: Width = 7 m, Length = 18 m
🧠 Example 2: Money & Age Word Problem
Problem:
A father is 24 years older than his son. After 2 years, the product of their ages will be 560. Find their current ages.
Solution:
Let son’s age = ‘x’ years
Father’s age = ‘x+24’ years
In 2 years:
Son’s age = x+2 & Father’s age = x+26
So, (x+2)(x+26)=560 → x²+28x+52 = 560 → x²+28x−508 = 0
Using the quadratic formula is much prudent than applying mid-term splitting because of a huge value of the constant term:
Answer: Son is 10 years old, Father is 34 years old
🧠 Example 3: Physics Application—Projectile Motion
Problem:
A ball is thrown upwards with an initial velocity such that its height after ttt seconds is given by h= −5t² + 20t. When does the ball hit the ground?
Solution:
Set h=0 i.e. −5t²+20t=0
Factor: −5t(t−4)=0 → t=0 (initial), or t=4
Answer: The ball hits the ground after 4 seconds.
🧠 Example 4: Revenue & Optimization
Problem:
A school charges $50 per student if 100 students attend. For every $1 increase in fee, 2 fewer students attend. What fee maximizes revenue?
Solution:
Let x = number of $1 increases
Fee = $ ’50+x’
Students = 100−2x
Revenue R=(50+x)(100−2x) → R=5000−100x+100x−2x²=5000−2x²
Maximum revenue at the vertex:
Oops! Recalculate correctly.
→ Expand: R=5000+100x−2x²−2x²
Actually, it’s
R=5000+100x−2x²
Fee = 50+25 = 75
Students = 100−2×25 = 50
Answer: Maximum revenue of $3750 at $75 per student
🧠 Example 5: Complex Quadratic Problem Involving Roots
Problem:
The sum of the reciprocals of two consecutive positive integers is 5/6. Find the integers.
Solution:
Let the smaller integer be x, next is x+1
Then:
Cross-multiply:
6(2x+1)=5x(x+1)
→ 12x+6 = 5x²+5x → 5x²−7x−6=0
Use the quadratic formula:
→ x=2 or x=−0.6
Answer: The integers are 2 and 3
📥 Downloadable Worksheet: 20 Quadratic Equation Word Problems
Practice is the key to mastering word problems. We’ve created a downloadable worksheet with 20 real-world quadratic problems, ranging from beginner to advanced levels. Each question has an online answer key (not detailed solutions) so you can self-evaluate after attempting.
👉 Click here to download the worksheet of Quadratic Equations Word Problems with answers (PDF)
🎉 Conclusion: Let’s Solve Quadratic Equations Word Problems Together!
Mastering quadratic equations word problems is a powerful skill that boosts confidence across all math topics. From calculating areas and ages to predicting optimal profits, these problems enhance both logic and application-based thinking.
At My Maths Club, we specialize in turning such challenging concepts into easy-to-grasp lessons through live, interactive classes for IGCSE students. Our expert-led sessions, free eBooks, AI-generated worksheets, and personalized attention make learning enjoyable and effective.
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