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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.

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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

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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

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Important Formulas

Standard Form

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

Degree of Quadratic Equation

Highest power of variable = 2

Area of Rectangle

$$Area=Length\times Breadth$$

Distance Formula

$$Distance=Speed\times Time$$

Consecutive Numbers

If first number is$$x$$

then second number is$$x+1$$

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Exercise 4.1 Solutions

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Question 1

Check whether the following are quadratic equations.

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(i)

$$(x+1)^2=2(x-3)$$

Solution

(ii)

$$x^2-2x=(-2)(3-x)$$

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(iii)

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

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(iv)

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

(v)

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

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(vi)

$$x^2+3x+1=(x-2)^2$$

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(vii)

$$(x+2)^3=2x(x^2-1)$$

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(viii)

$$x^3-4x^2-x+1=(x-2)^3$$

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Question 2

Represent the following situations in the form of quadratic equations.

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(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

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

Solution

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(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

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(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 $x^2$x2 is quadratic without simplification.
• Missing sign changes while shifting terms.
• Writing incorrect expressions in word problems.

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Exam Tips

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

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Practice MCQs

1. A quadratic equation has degree

A. 1

B. 2

C. 3

D. 4

✅ Answer: B

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2. Which is the standard form?

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

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

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

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

✅ Answer: B

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3. Which is NOT a quadratic equation?

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

B. $2x^2+3x=1$2×2+3x=1

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

D. $4x^2-9=0$4×2−9=0

✅ Answer: C

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4. Highest power in a quadratic equation is

A. 1

B. 2

C. 4

D. 5

✅ Answer: B

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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:$$ax^2+bx+c=0,\ a\neq0$$

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Q2. Why should equations be simplified first?

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

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Q3. Can a quadratic equation have only one solution?

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

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Q4. What is the standard form?

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

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Q5. Which chapter follows Exercise 4.1?

Exercise 4.2, where students solve quadratic equations using factorisation.

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📘 Found these Exercise 4.1 solutions helpful?

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