quadratic equation class 10 assignment pdf

NCERT Solutions Class 10 Maths Chapter 4 Quadratic Equations

NCERT solutions for class 10 maths chapter 4 Quadratic Equations deal with the concept of quadratic equations and the different ways of finding their roots. A quadratic equation is represented as ax 2 + bx + c = 0 , where a, b, c are the values of real numbers, and the value of ‘a’ is not equal to zero. This is also known as the standard form of the quadratic equation. An interesting fact to note is that many people believe that Babylonians were the first to solve quadratic equations. For instance, they knew how to find two positive numbers with a given positive sum and a given positive product, and this problem is equivalent to solving a quadratic equation. Moreover, the Greek mathematician Euclid developed a geometrical approach for finding out lengths which, in our present-day terminology, are solutions of quadratic equations The NCERT solutions class 10 maths chapter 4 Quadratic Equations teaches kids how to solve these equations by the factorization method and completing the square method. Students will come across important formulas like the quadratic formula for finding the roots of the equation.

The major takeaways from this chapter would be that for a quadratic equation two real roots will be distinct, if b 2 – 4ac > 0 ; two coincident roots will be obtained, if b 2 – 4ac = 0 ; and no roots will be there, if b 2 – 4ac < 0. The students will also explore how quadratic equations can be applied to real-life situations. The pdf file of the class 10 maths NCERT Solutions Chapter 4 Quadratic Equations in detail can be found below and also you can find some of these in the exercises given below.

• NCERT Solutions Class 10 Maths Chapter 4 Ex 4.1
• NCERT Solutions Class 10 Maths Chapter 4 Ex 4.2
• NCERT Solutions Class 10 Maths Chapter 4 Ex 4.3
• NCERT Solutions Class 10 Maths Chapter 4 Ex 4.4

NCERT Solutions for Class 10 Maths Chapter 4 PDF

The quadratic formula was discovered in 1025 AD by Sridharacharya. Since then, the quadratic equations have been widely used not just in mathematics but for real-life situations as well. This chapter explains the concept of solving quadratic equations with the help of real world practical examples so as to keep the learning process engaging and interesting. Students can have a quick glance of each section in the NCERT solutions for class 10 maths chapter 4 Quadratic Equations from the below-mentioned links :

☛ Download Class 10 Maths NCERT Solutions Chapter 4 Quadratic Equations

NCERT Class 10 Maths Chapter 4   Download PDF

NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations

Owing to the applicability of quadratic equations in various real-life problems, it becomes an important topic that needs to be understood well. The chapter showcases enough solved examples to ensure the coverage of the topic properly. Students will learn the basics of the various methods involved in finding the roots of quadratic equations along with the graphical representation of the equation. A brief exercise-wise analysis of NCERT Solutions Class 10 Maths Chapter 4 Quadratic Equations can be seen below :

• Class 10 Maths Chapter 4 Ex 4.1 - 2 Questions
• Class 10 Maths Chapter 4 Ex 4.2 - 6 Questions
• Class 10 Maths Chapter 4 Ex 4.3 - 11 Questions
• Class 10 Maths Chapter 4 Ex 4.4 - 5 Questions

Topics Covered: The topics covered in the class 10 maths NCERT Solutions Chapter 4 Quadratic Equations are the definition of quadratic equations , standard form of a quadratic equation, nature of roots , the concept of discriminant, quadratic formula, solution of a quadratic equation by the factorization method , and completing the square method.

Total Questions: Class 10 maths chapter 4 Quadratic Equations consists of a total of 24 questions, out of which 15 are straightforward, 5 are intermediate level questions, and 4 are difficult problems. These questions are explained in a step-wise manner. The important points are written in lucid language to encourage better comprehension.

List of Formulas in NCERT Solutions Class 10 Maths Chapter 4

NCERT solutions for class 10 maths chapter 4 will help students deal with different methods of solving a quadratic equation, and these methods involve the use of various formulas, which have been explained step by step in the chapter. Along with the formulas, kids also need to understand their implications. For example, as mentioned above the value of the discriminant can help us gain important insights into the practical applications of that particular quadratic equation. It is a good idea to make a formula sheet that will give kids a quick overview of these points as and when required. A few important formulas are as given below :

• Standard form of a quadratic equation: ax² + bx + c = 0, a ≠ 0
• Quadratic formula : [-b±√(b²-4ac)]/(2a) to find the solution of a quadratic equation.
• Discriminant : b 2 – 4ac

Important Questions for Class 10 Maths NCERT Solutions Chapter 4

Video Solutions for Class 10 Maths NCERT Chapter 4

FAQs on NCERT Solutions Class 10 Maths Chapter 4 Quadratic Equations

Why are ncert solutions class 10 maths chapter 4 vital for scoring well.

The NCERT solutions class 10 maths chapter 4 has been designed by scholars in their respective fields. After a lot of research, they have compiled the content into this textbook, making it a precious resource for study. The quadratic equations have widespread applications in our daily lives; hence, to understand them properly, it is advisable for the students to go through this chapter thoroughly. Also, the CBSE board recommends following these solutions, thereby making them of utmost importance.

Do I Need to Practice all Questions Given in Class 10 Maths NCERT Solutions Quadratic Equations?

The NCERT Solutions Class 10 Maths Quadratic Equations covers a wide range of topics that are important from the board exam's perspective. Problems based on topics such as the roots of a quadratic equation and the factorization method are often asked in exams. Hence, it is worthwhile to go through the entire theory and the solved examples from this chapter to get a better grasp of the concepts.

What are the Important Topics Covered in NCERT Solutions Class 10 Maths Chapter 4?

The important topics covered in the NCERT Solutions Class 10 Maths Chapter 4 are how to represent the given problem statements mathematically, what is the standard form of a quadratic equation, how to solve quadratic equations by factoring and completing the squares which is an essential topic requiring regular practice. Solving questions related to these topics will help the students score well in their board exams.

How Many Questions are there in NCERT Solutions Class 10 Maths Chapter 4?

The NCERT Solutions Class 10 Maths Chapter 4 contains a total of 24 well-researched questions by the subject experts. These 24 problems include both theoretical as well as graph-based questions. All the solutions are elaborated well in the NCERT solutions and are self-explanatory. The questions are solved in more than one method to strengthen the understanding of basic quadratic equation concepts.

How CBSE Students can utilize NCERT Solutions Class 10 Maths Chapter 4 effectively?

Students can utilize the NCERT Solutions Class 10 Maths Chapter 4 effectively by reviewing the principles and theorems explained in each section of the lesson on a frequent basis. They must then solve the exercise questions after practicing all of the examples and reviewing fundamental formulas related to factorization as well as completion of squares method of quadratic equations. This will help them build up their problem-solving approach thereby building confidence in maths.

Why Should I Practice NCERT Solutions Class 10 Maths Chapter 4?

Quadratic equations is one of the topics that is not just important for mathematics but also plays a fundamental role in a lot of real-life scenarios. If you have to calculate the length and breadth of a garden, you can use the quadratic equation. Based on this information, you can plan the quantity of a grass carpet needed for the garden. Quadratic equations are also used in fields such as astronomy, science, architecture. Owing to its vast range of applications, practicing the NCERT Solutions Class 10 Maths Chapter 4 in detail becomes very important for the students. Thus, to perfect this lesson practice is key.

NCERT Solutions for Class 10 Maths Chapter 4 – Quadratic Equations

Ncert solutions for class 10 maths chapter 4 – quadratic equations pdf.

Free PDF of NCERT Solutions for Class 10 Maths Chapter 4 – Quadratic Equations includes all the questions provided in NCERT Books prepared by Mathematics expert teachers as per CBSE NCERT guidelines from Mathongo.com. To download our free pdf of Chapter 4 Quadratic Equations Maths NCERT Solutions for Class 10 to help you to score more marks in your board exams and as well as competitive exams.

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Class 10 Mathematics Quadratic Equation Worksheets

We have provided below free printable  Class 10 Mathematics Quadratic Equation Worksheets  for Download in PDF. The worksheets have been designed based on the latest  NCERT Book for Class 10 Mathematics Quadratic Equation . These  Worksheets for Grade 10 Mathematics Quadratic Equation  cover all important topics which can come in your standard 10 tests and examinations.  Free printable worksheets for CBSE Class 10 Mathematics Quadratic Equation , school and class assignments, and practice test papers have been designed by our highly experienced class 10 faculty. You can free download CBSE NCERT printable worksheets for Mathematics Quadratic Equation Class 10 with solutions and answers. All worksheets and test sheets have been prepared by expert teachers as per the latest Syllabus in Mathematics Quadratic Equation Class 10. Students can click on the links below and download all Pdf  worksheets for Mathematics Quadratic Equation class 10  for free. All latest Kendriya Vidyalaya  Class 10 Mathematics Quadratic Equation Worksheets  with Answers and test papers are given below.

Mathematics Quadratic Equation Class 10 Worksheets Pdf Download

Here we have the biggest database of free  CBSE NCERT KVS  Worksheets for Class 10  Mathematics Quadratic Equation . You can download all free Mathematics Quadratic Equation worksheets in Pdf for standard 10th. Our teachers have covered  Class 10 important questions and answers  for Mathematics Quadratic Equation as per the latest curriculum for the current academic year. All test sheets question banks for Class 10 Mathematics Quadratic Equation and  CBSE Worksheets for Mathematics Quadratic Equation Class 10  will be really useful for Class 10 students to properly prepare for the upcoming tests and examinations. Class 10th students are advised to free download in Pdf all printable workbooks given below.

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Home » 10th Class » NCERT Book Class 10 Maths Chapter 4 Quadratic Equations (PDF)

NCERT Book Class 10 Maths Chapter 4 Quadratic Equations (PDF)

NCERT Book Class 10 Maths Chapter 4 Quadratic Equations is here. You can read and download Class 10 Maths Chapter 4 PDF from this page of aglasem.com. Quadratic Equations is one of the many lessons in NCERT Book Class 10 Maths in the new , updated version of 2023-24 . So if you are in 10th standard , and studying Maths textbook (named Mathematics ), then you can read Ch 4 here and afterwards use NCERT Solutions to solve questions answers of Quadratic Equations.

NCERT Book Class 10 Maths Chapter 4 Quadratic Equations

The complete Chapter 4 , which is Quadratic Equations , from NCERT Books for Class 10 Maths is as follows.

NCERT Book Class 10 Maths Chapter 4 Quadratic Equations PDF Download Link – Click Here To Download The Complete Chapter PDF

NCERT Book Class 10 Maths Full Book PDF Download Link – Click Here To Download The Complete Book PDF

NCERT Book Class 10 Maths Chapter 4 Quadratic Equations PDF

The direct link to download class 10 Maths NCERT Book PDF for chapter 4 Quadratic Equations is given above. However if you want to read the complete lesson on Quadratic Equations then that is also possible here at aglasem. So here is the complete class 10 Maths Ch 4 Quadratic Equations.

NCERT Book for Class 10 Maths

Besides the chapter on Quadratic Equations, you can read or download the NCERT Class 10 Maths PDF full book from aglasem. Here is the complete book:

• Chapter 1: Real Numbers
• Chapter 2: Polynomials
• Chapter 3: Pair Of Linear Equations In Two Variables
• Chapter 4: Quadratic Equations
• Chapter 5: Arithmetic Progressions
• Chapter 6: Triangles
• Chapter 7: Coordinate Geometry
• Chapter 8: Introduction to Trigonometry
• Chapter 9: Some Applications of Trigonometry
• Chapter 10: Circles
• Chapter 11: Area related to circles
• Chapter 12: Surface Areas And Volumes
• Chapter 13: Statistics
• Chapter 14: Probability
• NCERT Books for Class 10

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• NCERT Book Class 10 English
• NCERT Book Class 10 Hindi
• NCERT Book Class 10 Maths
• NCERT Book Class 10 Sanskrit
• NCERT Book Class 10 Science
• NCERT Book Class 10 Social Science

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Class 10 Maths Chapter 4 Quadratic Equations NCERT Textbook – An Overview

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Quadratic Equations Class 10 Notes CBSE Maths Chapter 4 (Free PDF Download)

• Revision Notes
• Chapter 4 Quadratic Equations

Class 10 Maths Revision Notes for Quadratic Equations of Chapter 4 - Free PDF Download

CBSE Class 10 Maths Notes Chapter 4 curated by Vedantu are most preferred by students during their exam preparation. The Class 10 Maths Chapter 4 Notes come with shortcut techniques along with step by step explanations of all topics. Quadratic Equations Class 10 Notes curated by subject experts are available as PDF downloads. The Quadratic Equations Notes are as per the syllabus of upcoming CBSE board exams. The study guides for all subjects of 10 th standard can be easily availed using the download option. Vedantu is a platform that also provides free NCERT Solutions and other study materials for students. You can download NCERT Solutions Class 10 Maths to help you to revise the complete Syllabus and score more marks in your examinations. Subjects like Science, Maths, English will become easy to study if you have access to NCERT Solution Class 10 Science , Maths solutions, and solutions of other subjects that are available on Vedantu only.

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Also, check CBSE Class 10 Maths revision notes for All chapters:

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Access Class 10 Maths Chapter 4-Quadratic Equation

Definition of the quadratic equation: .

A quadratic equation in the variable $x$ is an equation of the form $a{{x}^{2}}+bx+c=0$, where $a,b,c$ are real numbers, $a\ne 0$ . 

For example, $2{{x}^{2}}+x-300=0$ is a quadratic equation

The Standard Form of the Quadratic Equation:

Any equation of the form $p\left( x \right)=0$, where $p\left( x \right)$ is a polynomial of degree $2$ , is a quadratic equation. 

But when we write the terms of $p\left( x \right)$ in descending order of their degrees, then we get the standard form of the equation.

 That is, $a{{x}^{2}}+bx+c=0,a\ne 0$ is called the standard form of a quadratic equation.

Roots of the Quadratic Equation:

A solution of the equation $p\left( x \right)=a{{x}^{2}}+bx+c=0$, with $a\ne 0$ is called a root of the quadratic equation.

A real number $\alpha $  is called a root of the quadratic equation $a{{x}^{2}}+bx+c=0,a\ne 0$ if $a{{\alpha }^{2}}+b\alpha +c=0$.

It means $x=\alpha $ satisfies the quadratic equation or $x=\alpha $ is the root of the quadratic equation.

The zeroes of the quadratic polynomial $a{{x}^{2}}+bx+c$ and the roots of the quadratic equation $a{{x}^{2}}+bx+c=0$ are the same.

Method of Solving a Quadratic Equation:

Factorization Method

Factorize the quadratic equation by splitting the middle term.

After splitting the middle term, convert the equation into linear factors by taking common terms out.

Then on equating each factor to zero the roots are determined. 

For example: 

$\Rightarrow 2{{x}^{2}}-5x+3$            (Split the middle term)

$\Rightarrow 2{{x}^{2}}-2x-3x+3$    (Take out common terms to determine linear factors)

$\Rightarrow 2x\left( x-1 \right)-3\left( x-1 \right)$ 

$\Rightarrow \left( x-1 \right)\left( 2x-3 \right)$        (Equate to zero)

$\Rightarrow \left( x-1 \right)\left( 2x-3 \right)=0$

When $\left( x-1 \right)=0$ , $x=1$

When $\left( 2x-3 \right)=0$ , $x=\dfrac{3}{2}$ 

So, the roots of $2{{x}^{2}}-5x+3$ are $1$ and $\dfrac{3}{2}$ 

Method of completing the square

The solution of a quadratic equation can be found by converting any quadratic equation to perfect square of the form ${{\left( x+a \right)}^{2}}-{{b}^{2}}=0$.

To convert quadratic equation ${{x}^{2}}+ax+b=0$ to perfect square equate $b$ i.e., the constant term to the right side of equal sign then add square of half of $a$  i.e., square of half of coefficient of $x$  both sides.

To convert quadratic equation of form $a{{x}^{2}}+bx+c=0,a\ne 0$ to perfect square first divide the equation by $a$  i.e., the coefficient of ${{x}^{2}}$ then follow the above-mentioned steps.

For example:

$\Rightarrow {{x}^{2}}+4x-5=0$ (Equate constant term $5$  to the right of equal sign)

$\Rightarrow {{x}^{2}}+4x=5$      (Add square of half of $4$ both sides)

$\Rightarrow {{x}^{2}}+4x+{{\left( \dfrac{4}{2} \right)}^{2}}=5+{{\left( \dfrac{4}{2} \right)}^{2}}$

$ \Rightarrow {{x}^{2}}+4x+4=9$

$\Rightarrow {{\left( x+2 \right)}^{2}}=9 $

 $\Rightarrow {{\left( x+2 \right)}^{2}}-{{\left( 3 \right)}^{2}}=0 $

It is of the form ${{\left( x+a \right)}^{2}}-{{b}^{2}}=0$

$\Rightarrow {{\left( x+2 \right)}^{2}}-{{\left( 3 \right)}^{2}}=0$

$ \Rightarrow {{\left( x+2 \right)}^{2}}=9 $

$ \Rightarrow \left( x+2 \right)=\pm 3 $

      \[\Rightarrow x=1\] and $x=-5$  

So, the roots of ${{x}^{2}}+4x-5=0$are $1$ and $-5$ 

By using the quadratic formula

The root of a quadratic equation $a{{x}^{2}}+bx+c=0$ is given by formula 

$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ , where $\sqrt{{{b}^{2}}-4ac}$ is known as discriminant. 

If  $\sqrt{{{b}^{2}}-4ac}\ge 0$ then only the root of quadratic equation is given by 

$x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$

$\Rightarrow {{x}^{2}}+4x+3$ 

On using quadratic formula, we get

$ \Rightarrow x=\dfrac{-4\pm \sqrt{{{\left( 4 \right)}^{2}}-4\times 1\times 3}}{2\times 1} $

$ \Rightarrow x=\dfrac{-4\pm \sqrt{16-12}}{2}$

$ \Rightarrow x=\dfrac{-4\pm \sqrt{4}}{2}$

$ \Rightarrow x=\dfrac{-4\pm 2}{2} $

$\Rightarrow x=\dfrac{-4+2}{2}$ , $x=-1$ 

$\Rightarrow x=\dfrac{-4-2}{2}$ ,$x=-3$ 

So, the roots of ${{x}^{2}}+4x+3=0$ are $-1$  and $-3$ 

Nature of Roots Based on Discriminant:

If $\sqrt{{{b}^{2}}-4ac}=0$ then the roots are real and equal

If $\sqrt{{{b}^{2}}-4ac}>0$ then the roots are real and distinct 

If $\sqrt{{{b}^{2}}-4ac}<0$then the roots are imaginary

Quadratic Equation Notes – A Short Summary

Sridharacharya derived the quadratic formula in A.D 1025. This chapter is crucial not only from an exam point of view but also to deal with various real-life situations. For instance, you can calculate the length and breadth of a hall using quadratic equations. You can form an equation based on the information that the hall has a carpet area of 300 sq. metres and its length is one metre more than twice its breadth.

In this chapter, you will learn more about quadratic equations and how to find their roots. The Quadratic Equation Class 10 Notes for this chapter have been formulated by expert subject teachers to assist students with their overall preparation for the board exam. For this reason, the revision notes follow the NCERT guidelines to maintain accuracy and a high standard.

The relevant sub-topics under this chapter have been highlighted in our Quadratic Equations Notes to help students go through them quickly while revising. Key concepts and essential definitions come with an in-depth explanation to encourage better comprehension on your part. 

The revision notes are written in lucid language to boost your understanding and help you memorise them quickly.  Read through our revision notes to lend further clarity to your doubts and increase your chances of scoring high grades. Download the Quadratic Equation Class 10 Notes PDF and make them a part of your year-round study schedule.

Quadratic Equation Notes Class 10 – Revision Notes

Quadratic Equations Notes by us are divided into following sub-sections to help you revise efficiently before the examination.

Quadratic Equation Class 10 Notes – Roots of a Quadratic Equation

Under this section, you will learn to calculate the roots of quadratic equations. You should remember that quadratic equations in the variable x are an equation of the form ax 2 + bx + c = 0. Here a, b and c are real numbers. It has been explained in our Quadratic Equations Notes with the help of an example. 

As you go through our Quadratic Equations Notes, you will learn why this form of quadratic equations is called its standard form. Our revision notes will help you to revise how quadratic equations are used to represent real-life situations mathematically based on the information given in the question.

Under this section from our Class 10 Maths Chapter 4 Notes, you will also learn how to find out the roots of quadratic equations using shortcut techniques. While calculating the roots, you must remember that a root is a real number which satisfies the quadratic equation and is not equal to zero.

Quadratic Equation Notes Class 10 – Solution of Quadratic Equations By Factorisation

Quadratic equations can be solved by applying more than one method. In this class, you will learn to factorise quadratic equations to find its roots. However, it would help if you kept in mind that to factorise quadratic polynomials, you must split the middle term. You should also remember that to determine the roots; you must factorise the equation into linear factors and equate each factor to zero.

After solving the quadratic equation and finding its roots, you must verify that these are the roots of the equation given. You can go through our revision notes to review the method correctly. Since this is a very crucial method, you must practise it more than once and refer to our Quadratic Equations Notes to clarify your doubts, if any.

 You can even download our notes for Quadratic Equations Class 10 PDF and make it a part of your exam preparation process. Our notes contain a clear step by step explanation of the factorisation process, which you can easily memorise while revising a day before your board exams.

Class 10 Maths Chapter 4 Notes – Methods of Square

It is yet another method that can be used to solve a quadratic equation and find out its roots. Under this method, the quadratic equation ax 2 + bx + c = 0 is converted to the form (x + a) 2 – b 2 = 0. Now you have to apply your knowledge of square and square roots to solve this equation. Hence, this method is known as the method of completing the square. The terms containing x is completely inside a square, and the roots were found by taking the square roots.

This above method has been explained in our Quadratic Equations Notes with the help of an example. Students are given a quadratic equation x 2 + 4x. Now, employing this method, this will be converted to (x + 2) 2 – 4 = (x + 2) 2 - 2 2 . 

Under this section, you can also recall your knowledge of the quadratic formula. This formula is used to calculate the roots of a quadratic equation. Using this method, you can quickly solve a quadratic equation to find its roots.  So revise this technique again and again to increase your accuracy and speed. Referring to our Quadratic Equation PDF Class 10 will also help you to improve your understanding.

Quadratic Equations Class 10 Notes – Nature of Roots

You will learn about discriminant of a quadratic equation in the form ax 2 + bx + c = 0. Our Quadratic Equations Notes analyses this for your benefit. You can quickly revise the following pointers by downloading our revision notes.

 A quadratic equation ax 2 + bx + c = 0 has

Two distinct real roots, if b 2 – 4ac > 0.

Two equal real roots, if b 2 – 4ac = 0.

No real roots, if b 2 – 4ac < 0.

A clear grasp of this will help you to understand the nature of the roots. Read through our Quadratic Equations Class 10 Notes which has been framed by expert maths teachers to strengthen your understanding of the basic concepts. we have created a table that would let students know everything that they need to know about the nature of the roots of quadratic equations. That table is mentioned below.

The Relationship Between Roots of Quadratic Equations and Coefficients

It is important to understand the relationship between the roots of quadratic equations and coefficients while going through Ch 4 Maths Class 10 Notes. This is why we will go through that exact topic right now.

Let’s begin with the assumption that α and β are roots of a quadratic equation. This quadratic equation is ax 2 + bx + c. This means that:

α + β = -b/a

α – β = ±√[(α + β) 2 – 4αβ]

|α + β| = √D/|a|

Keep all of these equations in mind. From these equations, it can be said that the relationship between the roots and coefficients of a polynomial equation can be derived if one simplifies the given polynomials and substitutes the results. All of this can be depicted by the following equations:

α 2 β + β 2 α = αβ (α + β) = – bc/a 2

α 2 + αβ + β 2 = (α + β) 2 – αβ = (b 2 – ac)/a 2

α 2 + β 2 = (α – β) 2 – 2αβ

α 2 – β 2 = (α + β) (α – β)

α 3 + β 3 = (α + β) 3 + 3αβ(α + β)

α 3 – β 3 = (α – β) 3 + 3αβ(α – β)

(α/β) 2 + (β/α) 2 = α 4 + β 4 /α 2 β 2

The Range of Quadratic Equations

By now, we have looked at almost all the major formulas and theorems related to quadratic equations in these notes of Chapter 4 Maths Class 10. However, the one major topic that is still remaining is finding out the range of quadratic equations. So, let’s dive into this topic now.

Assume that a quadratic expression f (x) = ax 2 + bx + c. In this equation, a is not equal to 0. Also, a, b, and c are real. Hence, we can write the quadratic expression as:

F (x) = x 2 + bax = ca

Now, before moving forward, it is advised that you should look at the images that are given below.The first thing that you need to know about this image is that it shows the intervals in which the roots of a quadratic equation lie. We will look at all of these cases below.

Case 1: When both the roots of the quadratic equation are larger than any given number m

This happens when: b 2 - 4ac = (D) ≥ 0, -b / 2a > m, and f (m) > 0

Case 2: When both the roots of the quadratic expression are less than any given number ‘m’

This is true when b 2 - 4ac = (D) ≥ 0, -b / 2a < m, and f (m) > 0

Case 3: If both the roots of a quadratic equation lie in a particular interval (m 1 , m 2 )

This happens when b 2 - 4ac = (D) ≥ 0, m 1 < -b / 2a > m 2 , f (m 1 ) > 0, and f (m 2 ) > 0

Case 4: This happens when one root of a quadratic equation lies exactly in the given interval (m 1 , m 2 ) and f (m 1 ). f (m 2 ) < 0

Case 5: A given number ‘m’ will lie between the roots of a quadratic equation if f (m) < 0

Case 6: The roots of the quadratic equation will have opposite signs when f (0) < 0

Case 7: Both the roots of the quadratic equations are positive

This is true when b 2 - 4ac = (D) ≥ 0, α + β = -b / a > 0, and α x β = c / a > 0

Case 8: When both the roots of a quadratic equation are negative

This happens when b 2 - 4 ac = (D) ≥ 0, α + β = -b / a < 0, and α x β = c / a < 0

Fun Facts About Quadratic Equations

Did you know that you can use quadratic equations to solve many problems in the real world? This is true as quadratic equations can help you solve problems related to speed and geometry. Many problems related to quadrilateral figures and problems related to distance and time can also be solved by using quadratic equations.

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You can also easily access these notes such as Quadratic Equations Notes by downloading them in PDF format. Besides study guides, you can also enrol for our live tutorial classes via our online app. Our expert faculty members will guide you properly for thorough preparation.

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Other Maths Related Links

The following is a list of links for Maths-related topics that you can check out to improve your current understanding of Class 10 maths chapters and be able to score maximum marks in your exams.  

NCERT Solutions for Class 10

Revision Notes for Class 10

Important Questions for Class 10

CBSE Syllabus for Class 10

CSBE Sample Papers for Class 10

CSBE Previous Year Question Papers for Class 10

Maths formulas for Class 10

RD Sharma Class 10 Solutions

RS Aggarwal Solutions for Class 10

Lakhmir Singh Class 10 Solutions

NCERT Exemplar Class 10 Solutions

NCERT Books for Class 10

CBSE Class 10 Revision Notes - Other Chapters

The following are the links to the notes for all the important chapters in CBSE Class 10. We recommend that students visit the mentioned pages to get the most out of what Vedantu has to offer for students who wish to go the extra mile and prepare for their exams with the best materials out on the internet. 

Chapter 1 - Real Numbers Revision Notes

Chapter 2 - Polynomials Revision Notes

Chapter 3 - Pair of Linear Equations in Two Variables Revision Notes

Chapter 5 - Arithmetic Progressions Revision Notes

Chapter 6 - Triangles Revision Notes

Chapter 7 - Coordinate Geometry Revision Notes

Chapter 8 - Introduction to Trigonometry Revision Notes

Chapter 9 - Some Applications of Trigonometry Revision Notes

Chapter 10 - Circles Revision Notes

Chapter 11 - Constructions Revision Notes

Chapter 12 - Areas Related to Circles Revision Notes

Chapter 13 - Surface Areas and Volumes Revision Notes

Chapter 14 - Statistics Revision Notes

Chapter 15 - Probability Revision Notes

Quadratic Equations is a significant chapter in CBSE Class 10 Maths and our experts have worked diligently to come up with study materials and important notes to make sure students appearing for the Class 10 exam are prepared with all the necessary tools and information they need to score desirable marks. We recommend that students must go through these notes as well as the other links provided in this page to make the best out of these chapters.  

FAQs on Quadratic Equations Class 10 Notes CBSE Maths Chapter 4 (Free PDF Download)

1. What is the value of k for a quadratic equation (x - a) (x - 10) + 1 = 0. This equation has integral roots.

We can write the given quadratic equation as x 2 - (10 + k) x + 1 + 10k = 0

This means that, D = b 2 - 4ac = 100 + k 2 + 20k - 40k = k 2 - 20k + 96 = (k - 10) 2 - 4

This quadratic equation will have integral roots if the value of the discriminant > 0. This means that D is a perfect square and a = 1, while b and c are integers

Hence, (k - 10) 2 - D = 4

We know that the discriminant is a perfect square. This further means that the difference between the two perfect squares will be 4 only if D = 0 and (k - 10) 2 = 4

Therefore, k - 10 = ± 2. And the values of k = 8 and 12

2. What is the value of k when the equation p / (x + r) + q / (x - r) = k / 2x has two equal roots?

We can write the given quadratic equation as:

[2p + 2q - k] x 2 - 2r [p - q] x + r 2 k = 0

In case of equal roots, the discriminant or D = 0

This means that b 2 - 4ac = 0

In this equation, a = [2p + 2q - k], b = -2r [p - q], and c = r 2 k

                              = [-2r (p - q)] 2 - 4 (2p + 2q - k) (r 2 k)] = 0

                                  R 2 (p - q) 2 - r 2 k (2p + 2q - k) = 0

We also know that r is not equal to zero.

Hence, (p - q) 2 - k (2p + 2q - k) = 0

K 2 - 2(p + q) k + (p - q) 2

K = 2 (p + q) ± √[4 (p + q) 2 - 4 (p - q)] 2 / 2 = -(p + q) ± √4pq

Therefore, we can conclude that the value of k is (p + q) ± 2 √p x q = (√p ± √q) 2

NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations

Revised NCERT Solutions for class 10 Maths Chapter 4 Quadratic Equations in Hindi and English Medium updated for board exams 2024. The question answers and explanation of chapter 4 of 10th Maths are based on NCERT textbooks published for 2024-25. Class 10 Maths Chapter 4 Solutions for CBSE Board Class 10 Maths Chapter 4 Exercise 4.1 Class 10 Maths Chapter 4 Exercise 4.2 Class 10 Maths Chapter 4 Exercise 4.3 Class 10 Maths Chapter 4 Important Questions

Class 10 Maths Chapter 4 Solutions for State Boards Class 10 Maths Exercise 4.1 Class 10 Maths Exercise 4.2 Class 10 Maths Exercise 4.3 Class 10 Maths Exercise 4.4

Get the free Hindi Medium solutions for the academic session 2024-25. Download here UP Board Solutions for Class 10 Maths Chapter 4 all exercises. 10th Maths Chapter 4 solutions are online or download in PDF format. Download Assignments for practice with Solutions 10th Maths Chapter 4 Assignment 1 10th Maths Chapter 4 Assignment 2 10th Maths Chapter 4 Assignment 3 10th Maths Chapter 4 Assignment 4

10th Maths Chapter 4 NCERT Solutions follows the latest CBSE syllabus. Students of MP, UP, Gujarat board and CBSE can use it for Board exams. Class 10 Maths NCERT Solutions Offline Apps in Hindi or English for offline use. For any scholarly help, you may contact us. We will try to help you in the best possible ways.

NCERT Solutions for class 10 Maths Chapter 4 are given below in PDF format or view online. Solutions are in Hindi and English Medium. Uttar Pradesh students also can download UP Board Solutions for Class 10 Maths Chapter 4 here in Hindi Medium.

It is very essential to learn quadratic equations, because it have wide applications in other branches of mathematics, physics, in other subjects and also in real life situations. Download NCERT books 2024-25, revision books and solutions from the links given below.

Previous Year’s CBSE Questions

1. Two marks questions Find the roots of the quadratic equation √2 x² + 7x + 5√2 = 0. [CBSE 2017] 2. Find the value of k for which the equation x²+k(2x + k – 1) + 2 = 0 has real and equal roots. [CBSE 2017] 2. Three marks questions If the equation (1 + m² ) x² +2mcx + c² – a² = 0 has equal roots then show that c² = a² (1 + m²). 3. Four marks questions Speed of a boat in still water is 15 km/h. It goes 30 km upstream and returns back at the same point in 4 hours 30 minutes. Find the speed of the stream. [CBSE 2017]

The word quadratic is derived from the Latin word “Quadratum” which means “A square figure”. Brahmagupta (an ancient Indian Mathematician )(A.D. 598-665) gave an explicit formula to solve a quadratic equation. Later Sridharacharya (A.D. 1025) derived a formula, now known as the quadratic formula, for solving a quadratic equation by the method of completing the square. An Arab mathematician Al-khwarizni(about A.D. 800) also studied quadratic equations of different types. It is believed that Babylonians were the first to solve quadratic equations. Greek mathematician Euclid developed a geometrical approach for finding lengths, which are nothing but solutions of quadratic equations.

Important Questions on Class 10 Maths Chapter 4

Check whether the following is quadratic equation: (x + 1)² = 2(x – 3).

(x + 1)² = 2(x – 3) Simplifying the given equation, we get (x + 1)² = 2(x – 3) ⇒ x² + 2x + 1 = 2x – 6 ⇒ x² + 7 = 0 or x² + 0x + 7 = 0 This is an equation of type ax² + bx + c = 0. Hence, the given equation is a quadratic equation.

Represent the following situation in the form of quadratic equation: The product of two consecutive positive integers is 306. We need to find the integers.

Let the first integer = x Therefore, the second integer = x + 1 Hence, the product = x(x + 1) According to questions, x(x + 1) = 306 ⇒ x² + x = 306 ⇒ x² + x – 306 = 0 Hence, the two consecutive integers satisfies the quadratic equation x² + x – 306 = 0.

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.

Let, Shefali’s marks in Mathematics = x Therefore, Shefali’s marks in English = 30 – x If she got 2 marks more in Mathematics and 3 marks less in English, Marks in Mathematics = x + 2 Marks in English = 30 – x – 3 According to questions, Product = (x + 2)(27 – x) = 210 ⇒ 27x – x² + 54 – 2x = 210 ⇒(-x)² + 25x – 156 = 0 ⇒ x² – 25x + 156 = 0 ⇒ x² – 12x – 13x + 156 = 0 ⇒ x(x – 12) – 13(x – 12) = 0 ⇒ (x – 12)(x – 13) = 0 ⇒ (x – 12) = 0 or (x – 13) = 0 Either x = 12 or x = 13 If x = 12 then, marks in Maths = 12 and marks in English = 30 – 12 = 18 If x = 13 then, marks in Maths = 13 and marks in English = 30 – 13 = 1

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.

Let the larger number = x Let the smaller number = y Therefore, y² = 8x According to question, x² – y² = 180 ⇒ x² – 8x = 180 [As y² = 8x] ⇒ x² – 8x – 180 = 0 ⇒ x² – 18x + 10x – 180 = 0 ⇒ x(x – 18) + 10(x – 18) = 0 ⇒ (x – 18)(x + 10) = 0 ⇒ (x – 18) = 0 or (x + 10) = 0 Either x = 18 or x = -10 But x ≠ -10 , as x is the larger of two numbers. So, x = 18 Therefore, the larger number = 18 Hence, the smaller number = y = √144 = 12

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

Let the side of larger square = x m Let the side of smaller square = y m According to question, x² + y² = 468 …(i) Difference between perimeters, 4x – 4y = 24 ⇒ x – y = 6 ⇒ x = 6 + y … (ii) Putting the value of x in equation (i), we get (y + 6)² + y² = 468 ⇒ y² + 12y + 36 + y² = 468 ⇒ (2y)² +12y – 432 = 0 ⇒ y² + 6y – 216 = 0 ⇒ y² + 18y – 12y – 216 = 0 ⇒ y(y + 18) – 12(y + 18) = 0 ⇒ (y + 18)(y – 12) = 0 ⇒ (y + 18) = 0 or (y – 12) = 0 Either y = -18 or y = 12 But, y ≠ -18 , as x is the side of square, which can’t be negative. So, y = 12 Hence, the side of smaller square = 12 m Putting the value of y in equation (ii), we get Side of larger square = x = y + 6 = 12 + 6 = 18 m

How to Revise 10th Maths Chapter 4 Quadratic Equations for Exams

Schools and institutions across the world promptly acted according to a pandemic, by moving online. Tech advancement helped institutions transition physical classrooms to virtual ones in record time. For as long as I can remember, I have liked everything about mathematics – especially teaching young schoolers, I have seen some of the brilliant minds grapple to comprehend the concepts. I have seen hard-working bright-eyed students losing to perform better than average in the classroom. In this article, you will read some of the effective practices that helped many students score 100% in math of 10th standard chapter Quadratic equations.

Step 1: NCERT Solutions for Class 10 Maths Chapter 4 helps to practice real life based Problems in Exercise 4.3.

Step 2: class 10 maths chapter 4 solutions provides the fundamental facts of quadratic equations., step 3: ncert solutions class 10 maths chapter 4 by applying the perfect formula for answers., step 4: class 10 maths chapter 4 needs regular practice session after short intervals., step 5: practice class 10 maths chapter 4 from ncert textbook for exams..

How can I get good marks in Class 10 Maths Chapter 4 Quadratic Equations?

Student should know the methods of factorization to a quadratic equation. It will help a lot during the solution of questions in 10th Maths chapter 4. Quadratic formula is the ultimate trick to find the roots of difficult or easy format of any quadratic equation. So if someone has practiced well the factorization method and quadratic formula method, he will score better then ever in chapter 4 of class 10 mathematics.

How a quadratic polynomial is different from a quadratic equation in 10th Maths Chapter 4?

A polynomial of degree two is called a quadratic polynomial. When a quadratic polynomial is equated to zero, it is called a quadratic equation. A quadratic equation of the form ax² + bx + c = 0, a > 0, where a, b, c are constants and x is a variable is called a quadratic equation in the standard format.

In Class 10 Maths Chapter 4, which exercise is considered as the most difficult to solve?

Class 10 Maths, exercise 4.1, and 4.2 are easy to solve and having less number of questions. Exercise 4.3 is tricky to find the solutions and answers also. In this exercise most of the questions are based on application of quadratic equations.

What is meant by zeros of a quadratic equation in Chapter 4 of 10th Maths?

A zero of a polynomial is that real number, which when substituted for the variable makes the value of the polynomial zero. In case of a quadratic equation, the value of the variable for which LHS and RHS of the equation become equal is called a root or solution of the quadratic equation. There are three algebraic methods for finding the solution of a quadratic equation. These are (i) Factor Method (ii) Completing the square method and (iii) Using the Quadratic Formula.

What are the main topics to study in Class 10 Maths chapter 4?

In chapter 4 (Quadratic equations) of class 10th mathematics, Students will study

• 1) Meaning of Quadratic equations
• 2) Solution of a quadratic equation by factorization.
• 3) Solution of a quadratic equation by completing the square.
• 4) Solution of a quadratic equation using quadratic formula.
• 5) Nature of roots.

How many exercises are there in chapter 4 of Class 10th Maths?

There are in all 4 exercises in class 10 mathematics chapter 4 (Quadratic equations). In first exercise (Ex 4.1), there are only 2 questions (Q1 having 8 parts and Q2 having 4 parts). In second exercise (Ex 4.2), there are in all 6 questions. In fourth exercise (Ex 4.3), there are in all 5 questions. So, there are total 13 questions in class 10 mathematics chapter 4 (Quadratic equations). In this chapter there are in all 18 examples. Examples 1, 2 are based on Ex 4.1, Examples 3, 4, 5, 6 are based on Ex 4.2, Examples 16, 17, 18 are based on Ex 4.3.

Does chapter 4 of class 10th mathematics contain optional exercise?

No, chapter 4 (Quadratic equations) of class 10th mathematics doesn’t contain any optional exercise. All the four exercises are compulsory for the exams.

How much time required to complete chapter 4 of 10th Maths?

Students need maximum 3-4 days to complete chapter 4 (Quadratic equations) of class 10th mathematics. But even after this time, revision is compulsory to retain the way to solving questions.

« Chapter 3: Pair of Linear Equations in Two Variables

Chapter 5: arithmetic progression ».

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Assignment - Quadratic Equation, Class 10 Mathematics PDF Download

VERY SHORT ANSWER TYPE QUESTIONS

1. State which of the following equations are quadratic equations :

(i) 3x + 1/x – 8 = 0       (ii) 18x2 – 6x = 0     (iii) x 2 – 5x = 7 – 6x 3     (iv) x 2 = 25     (v) 6x 5 +3x 2 –7=0     (vi) x + 1/x 2 = 3 (vii) 5x 2 + 6x =7     (viii) 5x 3 – 2x – 3 = 0

2. Represent each of the following situations in the form of a quadratic equation : (i) The sum of the squares of two consecutive positive integers is 545. We need to find the integers. (ii) The hypotenuse of a right triangle is 25 cm. The difference between the lengths of the other two sides of the triangle is 5 cm. We need to find the lengths of these sides. (iii) One year ago, the father was 8 times as old as his son. Now his age is square of the son's age. We need to find their present ages. (iv) Ravi and Raj together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with. (v) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was Rs. 750. We would like to find out the number of toys produced on that day.

3. In each of the following determine whether the given values are the solutions of the given equation or not :

4. In each of the following, find the value of k for which the given value is a solution of the given equation :

(i) (x + 3) (2x – 3k) = 0 : x = 6

(ii) 3√7x 2 – 4x + k = 0 ; x =√7/3

5. Find the values of p and q for which x = 2/3 and  x = – 3 are the roots of the equation px 2 + 7x + q = 0.

SHORT ANSWER TYPE QUESTIONS

1. 27x 2 – 12 = 0

3. 16(x – 4) 2 = 9 (x + 3) 2

4.  x 2 – 300 = 0

5. x 2 + (a–b) x = ab

6. (3x+a) (3x+b)=ab

7. x 2 –1+ √2x+√2 =0

8. 3√7x 2 + 4x – √7= 0

9. √3y 2  + 11y + 6√3 = 0

10. abx 2 – (a 2 + b 2 )x + ab = 0

Find the roots of each of the following quadratic equations by the method of completing the squares

25. x 2 – 6x + 4 = 0

26. 2x 2 – 5x + 3 = 0

27. √5x 2 + 9x + 4 √5 = 0 28. (5z + 2a) (3z + 4b) = 8ab

29. 2 √2x 2 + 15x + 2 = 0

30. Find the solutions of 3x 2 – 2√6x + 2 = 0 by the method of completing the squares when

(i) x is a rational number

(ii) x is a real number

31. Find the solutions of 15x 2 + 3 = 17x, when (i) x is a rational number(ii) x is a real number.

32. Find the solutions of 5x 2 – 6x – 2 = 0, when (i) x is a rational number (ii) x is a real number

Find the roots of each of the following quadratic equations by using the quadratic formula

1. Equations in question No. (i), (ii), (iv), (vii), (ix), (xi),(xiii), (xiv), (xvi) and (xvii) are quadratic equations. 2. (i) x2 + x – 272 = 0, where x is the smaller integer. (ii) x 2 + 5x – 300 = 0, where x is the length of one side. (iii) x 2 – 8x + 7 = 0, where x (in years) is the present age of son. (iv) x 2 – 45x + 324 = 0, where x is the number of marbles with Ravi. (v) x 2 – 55x + 750 = 0, where x (in km/h) is the speed of the train. 3. (a) (i) Both are solution (ii) x = -√2 is a solution but x = -2√2 is not a solution. (iii) x = 1/2 is a solution but x = –1/2 is not a solution. (iv) Both are solution (b) (i) Both are solution (ii) Both are solution 4. (i) k = 4, (ii) k =  -√7

5. p = 3, q = – 6

1. 2/3 , – 2/3

2. 4, –8

4. 10√3, -10√3

5. b, – a

7. 1, √2

14. –a, – b

15. – 1

16. 12, – 2

18. -5/2, 3/2

19. 5, – 1

20. 6, 40/13

21. –10, – 1/5

22. – 4/3 , 1/8

23. 11/5, 5/8

24. 3, – 7/11

25. 3 ± √5

26. 1, 3/2 

29. No solution

30. (i) No solution (ii) √2/3

33. No solution

36. 1/√2

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Quadratic Equation Worksheet Class 10 PDF with Answers

These Quadratic Equation worksheet PDF can be helpful for both teachers and students. Teachers can track their student’s performance in the chapter Quadratic Equation. Students can easily identify their strong points and weak points by solving questions from the worksheet. Accordingly, students can work on both weak points and strong points. 

All students studying in CBSE class 10th, need to practise a lot of questions for the chapter Quadratic Equation. Students can easily practise questions from the Quadratic Equation problems worksheet PDF. By practising a lot of questions, students can improve their confidence level. With the help of confidence level, students can easily cover all the concepts included in the chapter Quadratic Equation. 

Quadratic Equation Worksheets with Solutions

Solutions is the written reply for all questions included in the worksheet. With the help of Quadratic Equation worksheets with solutions, students can solve all doubts regarding questions. Students can have deep learning in the chapter Quadratic Equation by solving all their doubts. By solving doubts, students can also score well in the chapter Quadratic Equation. 

Quadratic Equation Worksheet PDF

Worksheet is a sheet which includes many questions to solve for class 10th students. The Quadratic Equation worksheet PDF provides an opportunity for students to enhance their learning skills. Through these skills, students can easily score well in the chapter Quadratic Equation. Students can solve the portable document format (PDF) of the worksheet from their own comfort zone. 

How to Download the Quadratic Equation Worksheet PDF? 

To solve questions from the Quadratic Equation worksheet PDF, students can easily go through the given steps. Those steps are- 

• Open Selfstudys website.

• Bring the arrow towards CBSE which can be seen in the navigation bar. 

• Drop down menu will appear, select KVS NCERT CBSE Worksheet. 

• A new page will appear, select class 10th from the given list of classes. 
• Select Mathematics from the given list of subjects. Now click the chapter’s name that is Quadratic Equation. 

Features of the Quadratic Equation Worksheet PDF

Before starting to solve questions from the Quadratic Equation problems worksheet PDF, students need to know everything about the worksheet. Those features are- 

• Variety of questions are included:  The Quadratic Equation Maths Worksheet for Class 10 includes varieties of questions. Those varieties of questions are- one mark questions, two mark questions, three mark questions, etc. 
• Solutions are provided:  Doubts regarding each question can be easily solved through the solutions given. Through solving questions, a student's comprehensive skill can be increased. 
• All concepts are covered:  By solving questions from the Quadratic Equation problems worksheet, students can easily cover all the concepts included in the chapter.  
• Created by Expert:  These worksheets are personally created by the subject experts. These Quadratic Equation worksheet pdf are created with proper research. 
• Provides plenty of questions:  The Quadratic Equation worksheet provides plenty of questions to practise. Through good practice, students can get engaged in the learning process. 

Benefits of the Quadratic Equation Worksheet PDF

With the help of Quadratic Equation problems worksheet PDF, students can easily track their performance. This is the most crucial benefit, other than this there are more benefits. Those benefits are- 

• Builds a strong foundation:  Regular solving questions from the worksheet can help students to build a strong foundation. Through the strong foundation, students can score well in the chapter Quadratic Equation. 
• Improves speed and accuracy:  While solving questions from the chapter Quadratic Equation, students need to maintain the speed and accuracy. Speed and accuracy can be easily maintained and improved by solving questions from the Quadratic Equation worksheet PDF. 
• Acts as a guide:  Quadratic Equation worksheets with solutions acts as guide for both the teachers and students. Through the worksheet, teachers can guide their students according to the answers given by them. Students can also analyse themselves with the help of answers and can improve accordingly. 
• Enhances the learning process:  Regular solving of questions from the worksheet can help students enhance their learning process. According to the learning skills, students can easily understand all topics and concepts included in the chapter Quadratic Equation.  
• Improvisation of grades:  Regular solving of questions from the worksheet can help students to improve their marks and grades. With the help of good marks and good grades, students can select their desired field further. 

Tips to Score Good Marks in Quadratic Equation Worksheet

Students are requested to follow some tips to score good marks in the Quadratic Equation worksheet. Those tips are-

• Complete all the concepts:  First and the most crucial step is to understand all the concepts included in the chapter Quadratic Equation.  
• Practise questions:  Next step is to practise questions from the Quadratic Equation problems worksheet. Through this students can identify all types of questions: easy, moderate, difficult, etc.  
• Note down the mistakes:  After practising questions, students need to note down the wrong sums that have been done earlier. 
• Rectify the mistakes:  After noting down the mistakes, students need to rectify all the mistakes made. 
• Maintain a positive attitude:  Students are requested to maintain a positive attitude while solving worksheets. By maintaining a positive attitude, students can improve speed and accuracy while solving the worksheets. 
• Remain focused:  Students need to remain focused while solving questions from the Quadratic Equation problems worksheet pdf. As it helps students to solve the questions as fast as possible. 

When should a student start solving the Quadratic Equation Worksheet PDF?

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NCERT Exemplar Class 10 Maths Solutions for Chapter 4 - Quadratic Equations

Ncert exemplar solutions class 10 maths chapter 4 – free pdf download.

NCERT Exemplar Solutions Class 10 Maths Chapter 4 Quadratic Equations are provided here to help students prepare for the board exams. The exemplars have been prepared by subject experts in accordance with the latest CBSE syllabus (2023-2024) and are available in PDF, which can be downloaded easily.

Chapter 4 in Class 10 Maths is an important chapter for Class 10 students. Besides, there will be several questions based on this chapter, and students have to solve these problems using quadratic equations. Further, while dealing with this chapter, students will learn about different quadratic equations and how to find solutions by either using the factorisation method or the square method. To help students grasp the complete concepts of this chapter, free NCERT Exemplar for quadratic equations is provided here.  Click here to get exemplars for all chapters.

Students can go through the exemplar problems and solutions for Chapter 4 to understand the concepts introduced in this chapter, such as:

• Representing the given situations mathematically in the form of quadratic equations
• Solving quadratic equations with the help of the factorisation method
• Solving quadratic equations by completing the square
• Determining the nature of roots

Students can access the Class 10 Maths Chapter 4 NCERT Exemplar PDF below.

Download the PDF of NCERT Exemplar Solutions for Class 10 Maths Chapter 4 Quadratic Equations

Access Answers to NCERT Exemplar Class 10 Maths Chapter 4

Exercise 4.1.

Choose the correct answer from the given four options in the following questions:

1. Which of the following is a quadratic equation?

(A) x 2 + 2 x + 1 = (4 – x ) 2 + 3 (B) –2 x 2 = (5 – x )(2x-(2/5))

(C) ( k + 1) x 2 + (3/2) x = 7, where k = –1 (D) x 3 – x 2 = ( x – 1) 3

(D) x 3 – x 2 = ( x – 1) 3

Explanation:

The standard form of a quadratic equation is given by,

ax 2  + bx + c = 0, a ≠ 0

(A) Given, x 2  + 2x + 1 = (4 – x) 2  + 3

x 2  + 2x + 1 = 16 – 8x + x 2  + 3

10x – 18 = 0

which is not a quadratic equation.

(B) Given, -2x 2  = (5 – x) (2x – 2/5)

-2x 2  = 10x – 2x 2  – 2 +2/5x

52x – 10 = 0

(C) Given, (k + 1) x 2  + 3/2 x  = 7, where k = -1

(-1 + 1) x 2  + 3/2 x = 7

3x – 14 = 0

(D) Given, x 3  – x 2  = (x – 1) 3

x 3  – x 2  = x 3  – 3x 2  + 3x – 1

2x 2  – 3x + 1 = 0

which represents a quadratic equation.

2. Which of the following is not a quadratic equation?

(A) 2( x – 1) 2 = 4 x 2 – 2 x + 1 (B) 2 x – x 2 = x 2 + 5

(C) ( √ 2x + √ 3) 2 + x 2 = 3 x 2 − 5 x (D) ( x 2 + 2 x ) 2 = x 4 + 3 + 4 x 3

(D) ( x 2 + 2 x ) 2 = x 4 + 3 + 4 x 3

A quadratic equation is represented by the form,

(A) Given, 2(x – 1) 2  = 4x 2  – 2x + 1

2(x 2  – 2x + 1) = 4x 2  – 2x + 1

2x 2  + 2x – 1 = 0

which is a quadratic equation.

(B) Given, 2x – x 2  = x 2  + 5

2x 2  – 2x + 5 = 0

(C) Given, (√2x + √3) 2   = 3x 2  – 5x

2x 2 + 2√6x + 3  = 3x 2  – 5x

x 2 – (5 + 2√6)x – 3 = 0

(D) Given, (x 2  + 2x) 2  = x 4  + 3 + 4x 2

x 4  + 4x 3  + 4x 2  = x 4  + 3 + 4x 2

4x 3  – 3 = 0

which is a cubic equation and not a quadratic equation.

3. Which of the following equations has 2 as a root?

(A) x 2 – 4 x + 5 = 0 (B) x 2 + 3 x – 12 = 0

(C) 2 x 2 – 7 x + 6 = 0 (D) 3 x 2 – 6 x – 2 = 0

(C) 2 x 2 – 7 x + 6 = 0

If 2 is a root then substituting the value 2 in place of x should satisfy the equation.

x 2  – 4x + 5 = 0

(2) 2  – 4(2) + 5 = 1 ≠ 0

So, x = 2 is not a root of x 2  – 4x + 5 = 0

(B) Given, x 2  + 3x – 12 = 0

(2) 2  + 3(2) – 12 = -2 ≠ 0

So, x = 2 is not a root of x 2  + 3x – 12 = 0

(C) Given, 2x 2  – 7x + 6 = 0

2(2) 2  – 7(2) + 6 = 0

Here, x = 2 is a root of 2x 2  – 7x + 6 = 0

(D) Given, 3x 2  – 6x – 2 = 0

3(2) 2  – 6(2) – 2 = -2 ≠ 0

So, x = 2 is not a root of 3x 2  – 6x – 2 = 0

4. If ½ is a root of the equation x 2 + kx – 5/4 = 0, then the value of k is

(A) 2 (B) – 2

(C) ¼ (D) ½

If ½ is a root of the equation

x 2  + kx – 5/4 = 0 then, substituting the value of ½ in place of x should give us the value of k.

Given, x 2  + kx – 5/4 = 0 where, x = ½

(½) 2 + k (½) – (5/4) = 0

(k/2) = (5/4) – ¼

5. Which of the following equations has the sum of its roots as 3?

(A) 2x 2 – 3x + 6 = 0 (B) –x 2 + 3x – 3 = 0

(C) √2x 2 – 3/√2x+1=0 (D) 3x 2 – 3x + 3 = 0

(B) –x 2 + 3x – 3 = 0

The sum of the roots of a quadratic equation ax 2  + bx + c = 0, a ≠ 0 is given by,

Coefficient of x / coefficient of x 2 = – (b/a)

(A) Given, 2x 2  – 3x + 6 = 0

Sum of the roots = – b/a = -(-3/2) = 3/2

(B) Given, -x 2  + 3x – 3 = 0

Sum of the roots = – b/a = -(3/-1) = 3

(C) Given, √2x 2 – 3/√2x+1=0

2x 2 – 3x + √2 = 0

(D) Given, 3x 2  – 3x + 3 = 0

Sum of the roots = – b/a = -(-3/3) = 1

Exercise 4.2

1. State whether the following quadratic equations have two distinct real roots. Justify your answer.

• x 2 – 3x + 4 = 0
• 2x 2 + x – 1 = 0
• 2x 2 – 6x + 9/2 = 0
• 3x 2 – 4x + 1 = 0
• (x + 4) 2 – 8x = 0
• (x – √ 2) 2 – 2(x + 1) = 0
• √ 2 x 2 –(3/√2)x + 1/√2 = 0
• x (1 – x) – 2 = 0
• (x – 1) (x + 2) + 2 = 0
• (x + 1) (x – 2) + x = 0

The equation x 2  – 3x + 4 = 0 has no real roots.

D = b 2  – 4ac

= (-3) 2  – 4(1)(4)

= 9 – 16 < 0

Hence, the roots are imaginary.

The equation 2x 2  + x – 1 = 0 has two real and distinct roots.

= 1 2  – 4(2) (-1)

= 1 + 8 > 0

Hence, the roots are real and distinct.

The equation 2x 2 – 6x + (9/2) = 0 has real and equal roots.

= (-6) 2 – 4(2) (9/2)

= 36 – 36 = 0

Hence, the roots are real and equal.

The equation 3x 2  – 4x + 1 = 0 has two real and distinct roots.

= (-4) 2  – 4(3)(1)

= 16 – 12 > 0

The equation (x + 4) 2  – 8x = 0 has no real roots.

Simplifying the above equation,

x 2  + 8x + 16 – 8x = 0

x 2  + 16 = 0

=  (0) – 4(1) (16) < 0

The equation (x – √2) 2 – √2(x+1)=0 has two distinct and real roots.

x 2 – 2√2x + 2 – √2x – √2 = 0

x 2 – √2(2+1)x + (2 – √2) = 0

x 2 – 3√2x + (2 – √2) = 0

= (– 3√2) 2 – 4(1)(2 – √2)

= 18 – 8 + 4√2 > 0

The equation √2x 2 – 3x/√2 + ½ = 0 has two real and distinct roots.

= (- 3/√2) 2 – 4(√2) (½)

= (9/2) – 2√2 > 0

The equation x (1 – x) – 2 = 0 has no real roots.

x 2  – x + 2 = 0

=  (-1) 2  – 4(1)(2)

= 1 – 8 < 0

The equation (x – 1) (x + 2) + 2 = 0 has two real and distinct roots.

x 2  – x + 2x – 2 + 2 = 0

x 2  + x = 0

=  1 2  – 4(1)(0)

= 1 – 0 > 0

The equation (x + 1) (x – 2) + x = 0 has two real and distinct roots.

x 2  + x – 2x – 2 + x = 0

x 2  – 2 = 0

= (0) 2  – 4(1) (-2)

= 0 + 8 > 0

2. Write whether the following statements are true or false. Justify your answers.

• Every quadratic equation has exactly one root.
• Every quadratic equation has at least one real root.
• Every quadratic equation has at least two roots.
• Every quadratic equations has at most two roots.
• If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
• If the coefficient of x2 and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots.

(i) False. For example, a quadratic equation  x 2  – 9 = 0 has two distinct roots – 3 and 3.

(ii) False. For example, equation  x 2  + 4 = 0 has no real root.

(iii) False. For example, a quadratic equation  x 2  – 4 x  + 4 = 0 has only one root which is 2.

(iv) True, because every quadratic polynomial has almost two roots.

(v) True, because in this case discriminant is always positive.

For example, in  ax 2 +  bx  +  c  = 0, as  a  and  c  have opposite sign,  ac  < 0

⟹ Discriminant =  b 2  – 4 ac  > 0.

(vi) True, because in this case discriminant is always negative.

For example, in  ax 2 +  bx  +  c  = 0, as  b  = 0, and  a  and  c  have same sign then  ac  > 0

⟹ Discriminant =  b 2  – 4 ac  = – 4  a c  < 0

3. A quadratic equation with integral coefficient has integral roots. Justify your answer.

No, a quadratic equation with integral coefficients may or may not have integral roots.

Justification

Consider the following equation,

8x 2  – 2x – 1 = 0

The roots of the given equation are ½ and – ¼ which are not integers.

Hence, a quadratic equation with integral coefficient might or might not have integral roots.

Exercise 4.3

1. Find the roots of the quadratic equations by using the quadratic formula in each of the following:

• 2 x 2 – 3 x – 5 = 0
• 5 x 2 + 13 x + 8 = 0
• –3 x 2 + 5 x + 12 = 0
• – x 2 + 7 x – 10 = 0
• x 2 + 2 √ 2 x – 6 = 0
• x 2 – 3 √ 5 x + 10 = 0
• (½)x 2 – √11x + 1 = 0

The quadratic formula for finding the roots of quadratic equation

ax 2 + bx + c = 0, a ≠ 0 is given by,

(i) 2 x 2 – 3 x – 5 = 0

(ii) 5 x 2 + 13 x + 8 = 0

(iii) –3 x 2 + 5 x + 12 = 0

(iv) – x 2 + 7 x – 10 = 0

(v) x 2 + 2 √ 2 x – 6 = 0

(vi) x 2 – 3 √ 5 x + 10 = 0

(vii) (½)x 2 – √11x + 1 = 0

Exercise 4.4

1. Find a natural number whose square diminished by 84 is equal to thrice of 8 more than the given number.

Let the natural number = ‘x’.

According to the question,

We get the equation,

x² – 84 = 3(x+8)

x² – 84 = 3x + 24

x² – 3x – 84 – 24 = 0

x² – 3x – 108 = 0

x² – 12x + 9x – 108 = 0

x(x – 12) + 9(x – 12) = 0

(x + 9) (x – 12)

⇒ x = -9 and x = 12

Since, natural numbers cannot be negative.

The number is 12.

2. A natural number, when increased by 12, equals 160 times its reciprocal. Find the number.

Let the natural number = x

When the number increased by 12 = x + 12

Reciprocal of the number = 1/x

According to the question, we have,

x + 12 = 160 times of reciprocal of x

x + 12 = 160/ x

x( x + 12 ) = 160

x 2 + 12x – 160 = 0

x 2 + 20x – 8x – 160 = 0

x( x + 20) – 8( x + 20)= 0

(x + 20) (x – 8) = 0

x + 20 = 0 or x – 8 = 0

x = – 20 or x = 8

The required number = x = 8

3. A train, travelling at a uniform speed for 360 km, would have taken 48 minutes less to travel the same distance if its speed were 5 km/h more. Find the original speed of the train.

Let original speed of train = x km/h

Time = distance/speed

Time taken by train = 360/x hour

And, Time taken by train its speed increase 5 km/h = 360/( x + 5)

It is given that,

Time taken by train in first – time taken by train in 2nd case = 48 min = 48/60 hour

360/x – 360/(x +5) = 48/60 = 4/5

360(1/x – 1/(x +5)) = 4/5

360 ×5/4 (5/(x² +5x)) =1

450 × 5 = x² + 5x

x² +5x -2250 = 0

x = (-5± √ (25+9000))/2

= (-5 ±√ (9025) )/2

= (-5 ± 95)/2

But x ≠ -50 because speed cannot be negative

So, x = 45 km/h

Hence, original speed of train = 45 km/h

4. If Zeba were younger by 5 years than what she really is, then the square of her age (in years) would have been 11 more than five times her actual age. What is her age now?

Let Zeba’s age = x

(x-5)²=11+5x

x²+25-10x=11+5x

x²-15x+14=0

x²-14x-x+14=0

x(x-14)-1(x-14)=0

x=1 or x=14

We have to neglect 1 as 5 years younger than 1 cannot happen.

Therefore, Zeba’s present age = 14 years.

BYJU’S provides online study materials such as notes, exemplar books, question papers, and Maths NCERT Solutions for Class 10 to help students prepare for board exams in the most efficient way and score good marks. All these materials are available in downloadable PDFs. Students can also get an idea of the question pattern and marking scheme of Chapter 4 in the board exam by solving the sample papers and previous years’ question papers .

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Unit 1: Algebra foundations

Unit 2: solving equations & inequalities, unit 3: working with units, unit 4: linear equations & graphs, unit 5: forms of linear equations, unit 6: systems of equations, unit 7: inequalities (systems & graphs), unit 8: functions, unit 9: sequences, unit 10: absolute value & piecewise functions, unit 11: exponents & radicals, unit 12: exponential growth & decay, unit 13: quadratics: multiplying & factoring, unit 14: quadratic functions & equations, unit 15: irrational numbers, unit 16: creativity in algebra.