NCERT Class 10 Maths Exercise 4.1 Solutions | Quadratic Equations

Short Introduction

Exercise 4.1 introduces students to Quadratic Equations, one of the most important topics in Class 10 Mathematics. In this exercise, students learn how to:

  • Identify whether an equation is quadratic.
  • Convert equations into standard form.
  • Represent practical situations mathematically using quadratic equations.

This exercise develops algebraic thinking and prepares students for solving quadratic equations in the next exercises.

Quick Information Box

Particular Details
Chapter 4
Chapter Name Quadratic Equations
Exercise 4.1
Board CBSE
Class 10
Topic Identifying Quadratic Equations & Forming Equations
Difficulty Level Easy to Moderate

Concepts Used (Topics Covered)

  • Definition of Quadratic Equation
  • Standard Form of Quadratic Equation
  • Degree of Polynomial
  • Expanding Algebraic Expressions
  • Simplifying Equations
  • Word Problems
  • Mathematical Modelling

Important Formulas

Standard Form

ax2+bx+c=0,a≠0ax^2+bx+c=0,\qquad a\neq0

Degree of Quadratic Equation

Highest power of variable = 2

Area of Rectangle

Area=Length×BreadthArea=Length\times Breadth

Distance Formula

Distance=Speed×TimeDistance=Speed\times Time

Consecutive Numbers

If first number isxx

then second number isx+1x+1

Exercise 4.1 Solutions

Question 1

Check whether the following are quadratic equations.

(i)

(x+1)2=2(x−3)(x+1)^2=2(x-3)

Solution

(ii)

x2−2x=(−2)(3−x)x^2-2x=(-2)(3-x)

(iii)

(x−2)(x+1)=(x−1)(x+3)(x-2)(x+1)=(x-1)(x+3)

(iv)

(x−3)(2x+1)=x(x+5)(x-3)(2x+1)=x(x+5)

(v)

(2x−1)(x−3)=(x+5)(x−1)(2x-1)(x-3)=(x+5)(x-1)

(vi)

x2+3x+1=(x−2)2x^2+3x+1=(x-2)^2

(vii)

(x+2)3=2x(x2−1)(x+2)^3=2x(x^2-1)

(viii)

x3−4x2−x+1=(x−2)3x^3-4x^2-x+1=(x-2)^3

Question 2

Represent the following situations in the form of quadratic equations.

(i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

Solution

(ii) The product of two consecutive positive integers is 306. We need to find the integers.

Solution

(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.

Solution

(iv) (iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Solution

Common Mistakes

  • Forgetting to convert the equation into standard form.
  • Not expanding brackets correctly.
  • Assuming every equation containing x2x^2x2 is quadratic without simplification.
  • Missing sign changes while shifting terms.
  • Writing incorrect expressions in word problems.

Exam Tips

  • Always simplify the equation first.
  • Check the highest power after simplification.
  • Ensure the coefficient of x2x^2x2 is not zero.
  • For word problems, define the variable clearly before forming the equation.
  • Write the final quadratic equation in standard form.

Practice MCQs

1. A quadratic equation has degree

A. 1

B. 2

C. 3

D. 4

✅ Answer: B

2. Which is the standard form?

A. ax+b=0ax+b=0ax+b=0

B. ax2+bx+c=0ax^2+bx+c=0ax2+bx+c=0

C. x3+x=0x^3+x=0x3+x=0

D. x+2=3x+2=3x+2=3

✅ Answer: B

3. Which is NOT a quadratic equation?

A. x2+5=0x^2+5=0x2+5=0

B. 2x2+3x=12x^2+3x=12×2+3x=1

C. x3+x=0x^3+x=0x3+x=0

D. 4x2−9=04x^2-9=04×2−9=0

✅ Answer: C

4. Highest power in a quadratic equation is

A. 1

B. 2

C. 4

D. 5

✅ Answer: B

Frequently Asked Questions (FAQ)

Q1. What is a quadratic equation?

A quadratic equation is an equation of degree 2 and is written in the form:ax2+bx+c=0, a≠0ax^2+bx+c=0,\ a\neq0

Q2. Why should equations be simplified first?

Some equations may appear quadratic or non-quadratic until simplified. Simplification helps identify the correct degree.

Q3. Can a quadratic equation have only one solution?

Yes. It can have two equal real roots when the discriminant is zero.

Q4. What is the standard form?

ax2+bx+c=0ax^2+bx+c=0

Q5. Which chapter follows Exercise 4.1?

Exercise 4.2, where students solve quadratic equations using factorisation.

📘 Found these Exercise 4.1 solutions helpful?

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