NCERT Solutions for Class 10 Mathematics Chapter 4 Quadratic Equations

NCERT Solutions for Class 10 Mathematics Chapter 4 - Quadratic Equations

Welcome to the Chapter 4 - Quadratic Equations, Class 10 Mathematics NCERT Solutions page. Here, we provide detailed question answers for Chapter 4 - Quadratic Equations. The page is designed to help students gain a thorough understanding of the concepts related to natural resources, their classification, and sustainable development.

Our solutions explain each answer in a simple and comprehensive way, making it easier for students to grasp key topics Quadratic Equations and excel in their exams. By going through these Quadratic Equations question answers, you can strengthen your foundation and improve your performance in Class 10 Mathematics. Whether you’re revising or preparing for tests, this chapter-wise guide will serve as an invaluable resource.

Exercise 4

Q 1:

Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

(i) 2x²– 3x + 5 = 0 (iii) 2x²– 6x + 3 = 0

(i) 2x² – 3x + 5 = 0

On comparing given equation ax² + bx + c = 0

We get,

a = 2, b = -3 and c = 5

We know discriminant (D) = b² – 4ac

D = (-3)² – 4 2 5

D = 9 – 40

D = -31

As, b² – 4ac < 0

Therefore, no real roots exist for the given equation.

Q 2:

Find the values of k for each of the following quadratic equations, so that they have two equal roots.
(i) 2x²+ kx + 3 = 0 (ii) kx (x – 2) + 6 = 0

Q 3:

Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m²? If so, find its length and breadth.

Q 4:

Is the following situation possible? If so, determine their present ages.
The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

Q 5:

Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.

Exercise 1

Q 1:

Check whether the following are quadratic equations :
(i) (x + 1)² = 2(x – 3) (ii) x² – 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) x²+ 3x + 1 = (x – 2)² (vii) (x + 2)³ = 2x (x2 – 1) (viii) x³ – 4x² – x + 1 = (x – 2)³

Q 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.

(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 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.

Exercise 2

Q 1:

Find the roots of the following quadratic equations by factorisation:

Q 2:

Solve the problems given in Example 1.

(i) let x and y are the no of marbles of Romil and Somi respectively.

Acc. to ques,

X + y = 45................... (1)

When they both lost their 5 marbles

(x – 5) (y – 5) = 124.................... (2)

From equation (1), we have

y = 45 – x

Put this value of x in equation (2), we get

(x – 5)(45 – x – 5) = 124

(x – 5)(40 – x) = 124

40x – x² – 200 + 5x = 128

x² - 45x + 324 = 0

x² - 36x - 9x + 324 = 0

x (x – 36) – 9 (x – 36) = 0

(x – 9) (x – 36) = 0

x = 9 or x = 36

Q 3:

Find two numbers whose sum is 27 and product is 182.

Q 4:

Find two consecutive positive integers, sum of whose squares is 365.

Q 5:

The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.

Q 6:

A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs 90, find the number of articles produced and the cost of each article.

Exercise 3

Q 1:

Find the roots of the following quadratic equations, if they exist, by the method of
completing the square:
(i) 2x²– 7x + 3 = 0 (ii) 2x²+ x – 4 = 0 (iv) 2x² + x + 4 = 0

Q 10:

An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11km/h more than that of the passenger train, find the average speed of the two trains.

Q 11:

Sum of the areas of two squares is 468 m². If the difference of their perimeters is 24 m, ind the sides of the two squares.

Q 2:

Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.

Q 3:

Find the roots of the following equations:

Q 4:

The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is 1/3. Find his present age.

Q 5:

In a class test, the sum of Shefali’s marks in Mathematics and English is 30. Had she got 2 marks more in Mathematics and 3 marks less in English, the product of their marks would have been 210. Find her marks in the two subjects.

Q 6:

The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

Let the shorter side of rectangular field be = x meter

The diagonal of the field = (x + 60) m

And, longer side is 30m more than shorter side = (x + 30) m

According to question;

Using Pythagoras theorem;

(x + 60)² = x² + (x + 30)²

x (x - 90) + 30 (x - 90) = 0

(x + 30) (x - 90) = 0

(x + 30) = 0 or (x - 90) = 0

Either, x = -30 or x = 90

But x ≠ -30 because length can never be negative.

Hence, shorter side of field is 90 m and longer side is (x + 30) = 120 m

Q 7:

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Q 8:

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the speed of the train.

Q 9:

Two water taps together can fill a tank in hours. The tap of larger diameter takes 10 hours less than the smaller one to fill the tank separately. Find the time in which each tap can separately fill the tank.

Frequently Asked Questions about Quadratic Equations - Class 10 Mathematics

- 1. How many questions are covered in Quadratic Equations solutions?
- All questions from Quadratic Equations are covered with detailed step-by-step solutions including exercise questions, additional questions, and examples.
- 2. Are the solutions for Quadratic Equations helpful for exam preparation?
- Yes, the solutions provide comprehensive explanations that help students understand concepts clearly and prepare effectively for both board and competitive exams.
- 3. Can I find solutions to all exercises in Quadratic Equations?
- Yes, we provide solutions to all exercises, examples, and additional questions from Quadratic Equations with detailed explanations.
- 4. How do these solutions help in understanding Quadratic Equations concepts?
- Our solutions break down complex problems into simple steps, provide clear explanations, and include relevant examples to help students grasp the concepts easily.
- 5. Are there any tips for studying Quadratic Equations effectively?
- Yes, practice regularly, understand the concepts before memorizing, solve additional problems, and refer to our step-by-step solutions for better understanding.

Exam Preparation Tips for Quadratic Equations

The Quadratic Equations is an important chapter of 10 Mathematics. This chapter’s important topics like Quadratic Equations are often featured in board exams. Practicing the question answers from this chapter will help you rank high in your board exams.

Latest Blog Posts

Stay updated with our latest educational content and study tips

Smart Questions to Ask in a Parent-Teacher Meeting | PTM Made Easy

Parent-Teacher Meetings (PTMs) are more than quick updates on marks — they’re a chance to build a real partnership between home and school. A good Parent-Teacher Meeting conversation helps parents see beyond grades. It opens up insights about a child’s strengths, struggles, emotions and even hidden talents. When parents participate actively, they don’t just track […]

The Secret to Smarter Learning — Building Strong Critical Thinking Skills

In today’s world of endless information , knowing how to think is more important than knowing what to think . From school projects to real – life decisions , critical thinking helps students question ideas , analyze facts and form logical conclusions . But what exactly does critical thinking mean ? Simply put , it’s […]

Study Smarter, Not Harder: Build Productive Habits That Stick

Every student dreams of better grades , stronger focus and more study time – but the real challenge isn’t starting, it’s staying consistent . Building productive study habits is not about studying all day , it’s about studying smart . In today’s fast – paced digital world, distractions are everywhere – from endless phone notifications […]

The Hidden Risks of Online Gaming for Children — Is your child safe while gaming online?

Online gaming has rapidly become one of the most popular pastimes among children. Whether it’s multiplayer mobile games , PC adventures or console challenges , kids are spending more time than ever in the virtual world . On the surface, gaming seems entertaining and even educational – improving hand- eye coordination , teamwork and problem […]

View All Blog Posts

Benefits of Using Our NCERT Solutions for Class

When it comes to excelling in your studies, having a well-structured study guide can make a huge difference. Our NCERT Solutions for Class provide you with a comprehensive, easy-to-understand, and exam-focused resource that is specifically tailored to help you maximize your potential. Here are some of the key benefits of using our NCERT solutions for effective learning and high scores:

NCERT Solutions for Effective Exam Preparation

Preparing for exams requires more than just reading through textbooks. It demands a structured approach to understanding concepts, solving problems, and revising thoroughly. Here’s how our NCERT solutions can enhance your exam preparation:

Clear Understanding of Concepts: Our NCERT solutions are designed to break down complex topics into simple, understandable language, making it easier for students to grasp essential concepts in . This helps in building a solid foundation for each chapter, which is crucial for scoring high marks.
Step-by-Step Solutions: Each solution is presented in a detailed, step-by-step manner. This approach not only helps you understand how to reach the answer but also equips you with the right techniques to tackle similar questions in exams.
Access to Important Questions: We provide a curated list of important questions and commonly asked questions in exams. By practicing these questions, you can familiarize yourself with the types of problems that are likely to appear in the exams and gain confidence in answering them.
Quick Revision Tool: Our NCERT solutions serve as an excellent tool for last-minute revision. The solutions cover all key points, definitions, and explanations, ensuring that you have everything you need to quickly review before exams.

Importance of Structured Answers for Scoring Higher Marks

In exams, it's not just about getting the right answer—it's also about presenting it in a well-structured and logical way. Our NCERT solutions for Class are designed to guide you on how to write answers that are organized and effective for scoring high marks.

Precise and Concise Answers: Our solutions are crafted to provide answers that are to the point, without unnecessary elaboration. This ensures that you don't waste time during exams and focus on delivering accurate answers that examiners appreciate.
Step-Wise Marks Distribution: We understand that exams often allot marks based on specific steps or points. Our NCERT solutions break down each answer into structured steps to ensure you cover all essential points required for full marks.
Improved Presentation Skills: By following the format of our NCERT solutions, you learn how to present your answers in a systematic and logical manner. This helps in making your answers easy to read and allows the examiner to quickly identify key points, resulting in better scores.
Alignment with NCERT Guidelines: Since exams are often set in alignment with NCERT guidelines, our solutions are tailored to follow the exact format and language that is expected in exams. This can improve your chances of scoring higher by meeting the examiner's expectations.