Applications of Quadratic Equations – Real Life Word Problems
Quadratic equations are one of the most important topics in Class 10 Algebra. While students often learn how to solve quadratic equations using factorization or the quadratic formula, the real challenge in exams comes from application-based word problems.
These problems convert real-life situations into mathematical equations. Understanding how to form a quadratic equation from a situation is a key skill tested in CBSE board exams.
What is a Quadratic Equation?
A quadratic equation is an equation of the form:
ax² + bx + c = 0
where:
- a, b, c are constants
- a ≠ 0
- x is the variable
Quadratic equations appear in many real-world problems such as area calculations, number problems, motion problems, and financial situations.
Steps to Solve Word Problems Using Quadratic Equations
- Understand the problem carefully
- Define the unknown variable
- Convert the statement into an equation
- Solve the quadratic equation
- Verify the solution with the original problem
Solved Quadratic Equation Word Problems
Problem 1: Product of Two Consecutive Numbers
The product of two consecutive positive integers is 132. Find the numbers.
Solution:
Let the first number be x.
Next consecutive number = x + 1
Product = 132
x(x + 1) = 132
x² + x − 132 = 0
Factorization:
x² + 12x − 11x − 132 = 0
x(x + 12) −11(x + 12) = 0
(x − 11)(x + 12) = 0
x = 11 or x = -12
Since numbers are positive:
Numbers = 11 and 12
Problem 2: Area of a Rectangle
The area of a rectangle is 60 m² and its length is 4 m more than its breadth. Find the dimensions.
Solution:
Let breadth = x
Length = x + 4
Area = Length × Breadth
x(x + 4) = 60
x² + 4x − 60 = 0
(x + 10)(x − 6) = 0
x = 6 or -10
Breadth = 6 m
Length = 10 m
Problem 3: Speed and Time Problem
A car travels 120 km at a certain speed. If the speed were 20 km/h faster, it would take 1 hour less. Find the original speed.
Solution:
Let original speed = x km/h
Time taken = 120/x
New speed = x + 20
New time = 120/(x+20)
According to problem:
120/x − 120/(x+20) = 1
Solving:
120(x+20) − 120x = x(x+20)
2400 = x² + 20x
x² + 20x − 2400 = 0
(x + 60)(x − 40) = 0
Speed = 40 km/h
Problem 4: Difference of Squares
The difference between the squares of two consecutive integers is 45. Find the integers.
Solution:
Let numbers be x and x + 1
(x+1)² − x² = 45
x² + 2x + 1 − x² = 45
2x + 1 = 45
x = 22
Numbers = 22 and 23
Problem 5: Area of a Square Garden
The area of a square garden is increased by 96 m² when its side is increased by 4 m. Find the original side.
Solution:
Let side = x
New side = x + 4
Increase in area:
(x+4)² − x² = 96
x² + 8x + 16 − x² = 96
8x + 16 = 96
8x = 80
x = 10
Original side = 10 m
Problem 6: Number Problem
The sum of two numbers is 27 and their product is 182. Find the numbers.
Solution:
Let one number be x
Other number = 27 − x
Product = 182
x(27 − x) = 182
27x − x² = 182
x² − 27x + 182 = 0
(x − 13)(x − 14) = 0
Numbers = 13 and 14
Where Quadratic Equations Are Used in Real Life
- Calculating projectile motion in physics
- Finding maximum or minimum area
- Engineering design problems
- Business profit optimization
- Architecture and construction planning
Exam Tips for Quadratic Word Problems
✔ Convert the statement into an equation step-by-step
✔ Simplify before solving
✔ Reject negative values when not applicable
Common Mistakes Students Make
- Not defining the variable clearly
- Incorrect translation of word statements
- Calculation errors while solving quadratic equation
- Choosing incorrect root from the solution
Final Revision Advice
Quadratic equation word problems are commonly asked in the CBSE Class 10 Mathematics board exam. Students should practice different types of problems including number problems, geometry problems, and motion problems. With regular practice, identifying the correct equation becomes easier.
