CBSE Class 10 Mathematics Chapter 4: Quadratic Equations NCERT Solutions

Reviewed by NCERT Help Editorial Team · Updated on 29 Jul 2026

NCERT Solutions PDF Class 10 PDF

This chapter delves into the fundamental concepts of Quadratic Equations for CBSE Class 10 Mathematics. Students will learn to identify whether a given equation is quadratic by simplifying it to the standard form ax^2 + bx + c = 0. The solutions also guide students on how to translate real-world problems involving areas, ages, and distances into quadratic equations. By working through these problems, students will develop a strong understanding of the definition of a quadratic equation and the ability to represent various scenarios mathematically. These NCERT Solutions are designed to aid students in mastering the initial steps of solving quadratic equations, crucial for their exam preparation and building a solid foundation in algebra.

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Board	CBSE
Class	Class 10
Subject	Mathematics
Session	2026
Language	English
Type	NCERT Solutions
Chapter	Chapter 4: Quadratic Equations

Chapter summary

Chapter 4, Quadratic Equations, focuses on understanding the standard form of a quadratic equation (ax^2 + bx + c = 0) and identifying if a given equation fits this form after simplification. It also covers the crucial skill of formulating quadratic equations from word problems, involving scenarios like areas of rectangles, consecutive integers, and age-related problems. The exercises in this chapter lay the groundwork for solving these equations in subsequent topics.

Learning outcomes

Understand the definition of a quadratic equation.
Simplify algebraic expressions to determine if they represent a quadratic equation.
Identify the standard form of a quadratic equation (ax^2 + bx + c = 0).
Formulate quadratic equations from given real-world situations.
Represent problems involving area, consecutive integers, and ages using quadratic equations.

Topics covered

Paper topics

Definition of Quadratic Equations
Standard form of Quadratic Equations (ax^2 + bx + c = 0)
Identifying Quadratic Equations
Simplification of Algebraic Expressions
Formulating Quadratic Equations from Word Problems
Area of Rectangular Plots
Consecutive Integers
Ages Problems
Distance, Speed, and Time Problems (Introduction)

Important topics

Identifying Quadratic Equations
Formulating Quadratic Equations from Word Problems
Standard form of Quadratic Equations
Simplification of Algebraic Expressions

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Questions and Solutions

Question 1

Check whether the following are quadratic equations:

$(x+1)^2 = 2(x-3)$
$x^2 - 2x = (-2)(3-x)$
$(x-2)(x+1) = (x-1)(x+3)$
$(x-3)(2x+1) = x(x+5)$
$(2x-1)(x-3)=(x+5)(x-1)$
$x^2+3x+1=(x-2)^2$
$(x+2)^3 = 2x(x^2-1)$
$x^3 - 4x^2 - x + 1 = (x-2)^3$

Solution:

To determine if an equation is quadratic, we need to simplify it to the standard form $ax^2 + bx + c = 0$ , where $a
eq 0$ .

Given equation: (x+1)^2 = 2(x-3)
Expand both sides: x^2 + 2x + 1 = 2x - 6
Rearrange terms to one side: x^2 + 2x - 2x + 1 + 6 = 0
Simplify: x^2 + 7 = 0
This equation is in the form ax^2 + bx + c = 0 with a=1, b=0, and c=7. Since a
eq 0, it is a quadratic equation.
Given equation: x^2 - 2x = (-2)(3-x)
Expand the right side: x^2 - 2x = -6 + 2x
Rearrange terms: x^2 - 2x - 2x + 6 = 0
Simplify: x^2 - 4x + 6 = 0
This equation is in the form ax^2 + bx + c = 0 with a=1, b=-4, and c=6. Since a
eq 0, it is a quadratic equation.
Given equation: $(x-2)(x+1) = (x-1)(x+3)$

Expand both sides: $x^2 + x - 2x - 2 = x^2 + 3x - x - 3$

Simplify each side: $x^2 - x - 2 = x^2 + 2x - 3$

Subtract $x^2$ from both sides: $-x - 2 = 2x - 3$

Rearrange terms: $-x - 2x - 2 + 3 = 0$

Simplify: $-3x + 1 = 0$ or $3x - 1 = 0$

This is a linear equation (highest power of x is 1), not a quadratic equation.
Given equation: (x-3)(2x+1) = x(x+5)
Expand both sides: 2x^2 + x - 6x - 3 = x^2 + 5x
Simplify the left side: 2x^2 - 5x - 3 = x^2 + 5x
Rearrange terms: 2x^2 - x^2 - 5x - 5x - 3 = 0
Simplify: x^2 - 10x - 3 = 0
This equation is in the form ax^2 + bx + c = 0 with a=1, b=-10, and c=-3. Since a
eq 0, it is a quadratic equation.
Given equation: (2x-1)(x-3) = (x+5)(x-1)
Expand both sides: 2x^2 - 6x - x + 3 = x^2 - x + 5x - 5
Simplify each side: 2x^2 - 7x + 3 = x^2 + 4x - 5
Rearrange terms: 2x^2 - x^2 - 7x - 4x + 3 + 5 = 0
Simplify: x^2 - 11x + 8 = 0
This equation is in the form ax^2 + bx + c = 0 with a=1, b=-11, and c=8. Since a
eq 0, it is a quadratic equation.
Given equation: $x^2 + 3x + 1 = (x-2)^2$

Expand the right side: $x^2 + 3x + 1 = x^2 - 4x + 4$

Subtract $x^2$ from both sides: $3x + 1 = -4x + 4$

Rearrange terms: $3x + 4x + 1 - 4 = 0$

Simplify: $7x - 3 = 0$

This is a linear equation, not a quadratic equation.
Given equation: $(x+2)^3 = 2x(x^2-1)$

Expand the left side using $(a+b)^3 = a^3 + b^3 + 3a^2b + 3ab^2$ : $x^3 + 2^3 + 3(x^2)(2) + 3(x)(2^2) = 2x^3 - 2x$

$x^3 + 8 + 6x^2 + 12x = 2x^3 - 2x$

Rearrange terms: $2x^3 - x^3 - 6x^2 - 2x - 12x - 8 = 0$

Simplify: $x^3 - 6x^2 - 14x - 8 = 0$

The highest power of x is 3, so this is a cubic equation, not a quadratic equation.
Given equation: x^3 - 4x^2 - x + 1 = (x-2)^3
Expand the right side using (a-b)^3 = a^3 - b^3 - 3a^2b + 3ab^2: x^3 - 4x^2 - x + 1 = x^3 - 2^3 - 3(x^2)(2) + 3(x)(2^2)
x^3 - 4x^2 - x + 1 = x^3 - 8 - 6x^2 + 12x
Rearrange terms: x^3 - x^3 - 4x^2 + 6x^2 - x - 12x + 1 + 8 = 0
Simplify: 2x^2 - 13x + 9 = 0
This equation is in the form ax^2 + bx + c = 0 with a=2, b=-13, and c=9. Since a
eq 0, it is a quadratic equation.

Question 2

Represent the following situations in the form of quadratic equations.

The area of a rectangular plot is 528 m<sup>2</sup>. 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.
The product of two consecutive positive integers is 306. We need to find the integers.
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.
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:

Let the breadth of the rectangular plot be $x$ metres.

According to the problem, the length of the plot is one more than twice its breadth. So, the length is $2x + 1$ metres.

The area of a rectangle is given by the formula: Area = Length $\times$ Breadth.

We are given that the area is 528 m<sup>2</sup>. Therefore,

$(2x + 1) \times x = 528$

Expanding this equation, we get:

$2x^2 + x = 528$

To express this in the standard quadratic form $ax^2 + bx + c = 0$ , we move all terms to one side:

$2x^2 + x - 528 = 0$

This is the quadratic equation representing the situation.
Let the first positive integer be $x$ .

Since the integers are consecutive positive integers, the next integer will be $x + 1$ .

The problem states that the product of these two consecutive positive integers is 306.

Therefore, we can write the equation as:

$x(x + 1) = 306$

Expanding the left side:

$x^2 + x = 306$

Rearranging the terms to form a standard quadratic equation $ax^2 + bx + c = 0$ :

$x^2 + x - 306 = 0$

This is the quadratic equation representing the situation.
Let Rohan's present age be $x$ years.

Rohan's mother is 26 years older than him, so her present age is $x + 26$ years.

In 3 years from now:

Rohan's age will be $x + 3$ years.

Rohan's mother's age will be $(x + 26) + 3 = x + 29$ years.

The problem states that the product of their ages 3 years from now will be 360.

So, we can write the equation as:

$(x + 3)(x + 29) = 360$

Expand the left side:

$x(x + 29) + 3(x + 29) = 360$

$x^2 + 29x + 3x + 87 = 360$

$x^2 + 32x + 87 = 360$

Rearrange the terms to form a standard quadratic equation $ax^2 + bx + c = 0$ :

$x^2 + 32x + 87 - 360 = 0$

$x^2 + 32x - 273 = 0$

This is the quadratic equation representing the situation.
Let the uniform speed of the train be $s$ km/h.

The distance to be traveled is 480 km.

The time taken to cover the distance at uniform speed is given by Time = Distance / Speed.

So, the original time taken is $t_1 = \frac{480}{s}$ hours.

According to the problem, if the speed had been 8 km/h less, the new speed would be $s - 8$ km/h.

The time taken to cover the same distance at this reduced speed would be $t_2 = \frac{480}{s - 8}$ hours.

The problem states that this new time ( $t_2$ ) is 3 hours more than the original time ( $t_1$ ).

Therefore, we can write the equation as:

$t_2 = t_1 + 3$

$\frac{480}{s - 8} = \frac{480}{s} + 3$

To solve this, we can first get a common denominator on the right side:

$\frac{480}{s - 8} = \frac{480 + 3s}{s}$

Now, cross-multiply:

$480s = (s - 8)(480 + 3s)$

Expand the right side:

$480s = s(480 + 3s) - 8(480 + 3s)$

$480s = 480s + 3s^2 - 3840 - 24s$

Subtract $480s$ from both sides:

$0 = 3s^2 - 24s - 3840$

Rearrange the terms to form a standard quadratic equation $as^2 + bs + c = 0$ :

$3s^2 - 24s - 3840 = 0$

We can simplify this equation by dividing all terms by 3:

$s^2 - 8s - 1280 = 0$

This is the quadratic equation representing the situation.

Common mistakes

Errors in algebraic expansion and simplification, leading to incorrect identification of quadratic equations.
Incorrectly setting up the relationship between variables in word problems.
Forgetting to simplify the equation to the standard form before concluding if it's quadratic.
Mistakes in transposing terms across the equality sign.

Revision tips

Practice simplifying each equation thoroughly to its standard form.
Pay close attention to the conditions given in word problems to correctly define variables and relationships.
Double-check your algebraic manipulations, especially when dealing with signs and exponents.
Ensure the final equation is in the form ax^2 + bx + c = 0 before classifying it.

Practice MCQs

Q1. Which of the following equations is a quadratic equation?

Q2. If the area of a rectangular plot is 528 $m^2$ and its length is one more than twice its breadth (x), what is the quadratic equation formed?

Q3. The product of two consecutive positive integers is 306. If the first integer is x, what is the equation?

Q4. Which equation is NOT quadratic after simplification?

Q5. The equation $x^3$ - 4 $x^2$ - x + 1 = (x-2)^3 simplifies to which quadratic form?

Frequently asked questions

What is the main goal of Chapter 4: Quadratic Equations in Class 10 Maths?

The primary goal is to understand what a quadratic equation is, how to identify one by simplifying it to the standard form ax^2 + bx + c = 0, and how to translate real-world situations into quadratic equations.

How do I check if an equation is quadratic?

You need to simplify the given equation by expanding terms and combining like terms. If the highest power of the variable is 2 and the coefficient of the x^2 term (a) is not zero, then it is a quadratic equation.

What kind of word problems are covered in this chapter?

This chapter covers word problems related to the area of rectangles, the product of consecutive integers, and problems involving ages. These situations are then represented as quadratic equations.

Are the solutions to the quadratic equations provided in Exercise 4.1?

Exercise 4.1 focuses on identifying and forming quadratic equations. The actual solving of these equations using methods like factorization or the quadratic formula is typically covered in subsequent exercises of the chapter.

Why is it important to represent situations as quadratic equations?

Representing situations as quadratic equations allows us to use mathematical tools to find unknown values that satisfy the given conditions. This is a fundamental skill in algebra and problem-solving.

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