---
title: "NCERT Solutions for Class 10 Maths Chapter 4 Exercise 4.1"
url: https://www.swavid.com/maths/class/10/chapter/quadratic-equations/ncert-solutions/exercise-4-1
dateModified: 2026-10-07T15:49:35+00:00
---

# NCERT Solutions for Class 10 Maths Chapter 4 Exercise 4.1

Chapter 4: Quadratic Equations. Every question from Exercise 4.1, with full working and the final answer.

Free PDF (10 pages): https://www.swavid.com/api/seo/pdf/ncert/maths/class-10/swavid-ncert-solutions-class-10-maths-chapter-4-quadratic-equations-fb40ded126.pdf

## EXERCISE 4.1

### Question 1

*3 marks · Short answer*

Check whether the following are quadratic equations :
(i) $(x + 1)^2 = 2(x - 3)$
(ii) $x^2 - 2x = (-2) (3 - x)$
(iii) $(x - 2)(x + 1) = (x - 1)(x + 3)$
(iv) $(x - 3)(2x + 1) = x(x + 5)$
(v) $(2x - 1)(x - 3) = (x + 5)(x - 1)$
(vi) $x^2 + 3x + 1 = (x - 2)^2$
(vii) $(x + 2)^3 = 2x (x^2 - 1)$
(viii) $x^3 - 4x^2 - x + 1 = (x - 2)^3$

**Part (i)**

1. Given equation: $(x + 1)^2 = 2(x - 3)$
2. Expanding both sides gives $x^2 + 2x + 1 = 2x - 6$
3. Simplifying yields $x^2 + 7 = 0$, which is of the form $ax^2 + bx + c = 0$.

Answer (i): Yes, it is a quadratic equation.

**Part (ii)**

1. Given equation: $x^2 - 2x = (-2)(3 - x)$
2. Expanding the right side gives $x^2 - 2x = -6 + 2x$
3. Rearranging terms yields $x^2 - 4x + 6 = 0$, which is of the form $ax^2 + bx + c = 0$.

Answer (ii): Yes, it is a quadratic equation.

**Part (iii)**

1. Given equation: $(x - 2)(x + 1) = (x - 1)(x + 3)$
2. Expanding both sides gives $x^2 - x - 2 = x^2 + 2x - 3$
3. Cancelling $x^2$ and simplifying yields $-3x + 1 = 0$, which is not of the form $ax^2 + bx + c = 0$.

Answer (iii): No, it is not a quadratic equation.

**Part (iv)**

1. Given equation: $(x - 3)(2x + 1) = x(x + 5)$
2. Expanding both sides gives $2x^2 - 5x - 3 = x^2 + 5x$
3. Rearranging terms yields $x^2 - 10x - 3 = 0$, which is of the form $ax^2 + bx + c = 0$.

Answer (iv): Yes, it is a quadratic equation.

**Part (v)**

1. Given equation: $(2x - 1)(x - 3) = (x + 5)(x - 1)$
2. Expanding both sides gives $2x^2 - 7x + 3 = x^2 + 4x - 5$
3. Rearranging terms yields $x^2 - 11x + 8 = 0$, which is of the form $ax^2 + bx + c = 0$.

Answer (v): Yes, it is a quadratic equation.

**Part (vi)**

1. Given equation: $x^2 + 3x + 1 = (x - 2)^2$
2. Expanding the right side gives $x^2 + 3x + 1 = x^2 - 4x + 4$
3. Cancelling $x^2$ and simplifying yields $7x - 3 = 0$, which is not of the form $ax^2 + bx + c = 0$.

Answer (vi): No, it is not a quadratic equation.

**Part (vii)**

1. Given equation: $(x + 2)^3 = 2x(x^2 - 1)$
2. Expanding both sides gives $x^3 + 6x^2 + 12x + 8 = 2x^3 - 2x$
3. Rearranging terms yields $-x^3 + 6x^2 + 14x + 8 = 0$, which has degree 3.

Answer (vii): No, it is not a quadratic equation.

**Part (viii)**

1. Given equation: $x^3 - 4x^2 - x + 1 = (x - 2)^3$
2. Expanding the right side gives $x^3 - 4x^2 - x + 1 = x^3 - 6x^2 + 12x - 8$
3. Cancelling $x^3$ and rearranging terms yields $2x^2 - 13x + 9 = 0$, which is of the form $ax^2 + bx + c = 0$.

Answer (viii): Yes, it is a quadratic equation.

**Answer:** Refer to the parts for individual results.

> Common mistake: Failing to expand binomial cubes or products completely before comparing the degree of the resulting equation.

### Question 2

*3 marks · Short answer*

Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is $528\text{ m}^2$. 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.
(ii) The product of two consecutive positive integers is $306$. We need to find the integers.
(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.
(iv) A train travels a distance of $480\text{ km}$ at a uniform speed. If the speed had been $8\text{ km/h}$ less, then it would have taken $3$ hours more to cover the same distance. We need to find the speed of the train.

**Part (i)**

1. Let the breadth of the plot be $x\text{ m}$.
2. Then the length of the plot is $(2x + 1)\text{ m}$.
3. Area = $\text{length} \times \text{breadth} = x(2x + 1) = 528$.
4. Simplifying gives $2x^2 + x - 528 = 0$.

Answer (i): $2x^2 + x - 528 = 0$

**Part (ii)**

1. Let the first positive integer be $x$.
2. Then the next consecutive positive integer is $x + 1$.
3. Their product is $x(x + 1) = 306$.
4. Simplifying gives $x^2 + x - 306 = 0$.

Answer (ii): $x^2 + x - 306 = 0$

**Part (iii)**

1. Let Rohan's present age be $x$ years.
2. Then Rohan's mother's present age is $(x + 26)$ years.
3. Three years from now, their ages will be $(x + 3)$ and $(x + 29)$ years respectively.
4. The product of their ages is $(x + 3)(x + 29) = 360$, which simplifies to $x^2 + 32x - 273 = 0$.

Answer (iii): $x^2 + 32x - 273 = 0$

**Part (iv)**

1. Let the uniform speed of the train be $x\text{ km/h}$.
2. Time taken to cover $480\text{ km}$ is $\frac{480}{x}\text{ hours}$.
3. If the speed is $(x - 8)\text{ km/h}$, the time taken is $\frac{480}{x - 8}\text{ hours}$.
4. According to the question, $\frac{480}{x - 8} - \frac{480}{x} = 3$, which simplifies to $x^2 - 8x - 1280 = 0$.

Answer (iv): $x^2 - 8x - 1280 = 0$

**Answer:** Refer to the parts for individual results.

> Common mistake: Formulating the algebraic expressions incorrectly or making errors in simplifying terms.

## Related pages

- [All Chapter 4 NCERT solutions](https://www.swavid.com/maths/class/10/chapter/quadratic-equations/ncert-solutions)
- [Exercise 4.2](https://www.swavid.com/maths/class/10/chapter/quadratic-equations/ncert-solutions/exercise-4-2)
- [Exercise 4.3](https://www.swavid.com/maths/class/10/chapter/quadratic-equations/ncert-solutions/exercise-4-3)

Solutions written by SwaVid, a personal AI tutor for Class 6 to 10 Maths and Science. Practise this chapter free: https://www.swavid.com/start/student