# NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations (Ex 4.1) Exercise 4.1

Understanding Quadratic Equations and several approaches for locating their Roots are covered in Chapter 4 of the Class 10 Maths NCERT Solutions.  Students in class 10 studying Maths chapter 4 will be able to fully comprehend linear equations at the root level.It thoroughly explains each key principle, assisting students in grasping the concepts.

The Maths Class 10 Chapter 4 Exercise 4.1 is an extremely important exercise for students. The section’s chapter 4 of the NCERT Solutions for Class 10 Maths focuses on key ideas including the Quadratic Equation definition, standard form of a Quadratic Equation, nature of Roots, Discriminant Concept, Quadratic Formula, Factorization Technique, and completing the Square Method. These answers are all created with the new CBSE pattern in mind to ensure that students fully comprehend their exams.

NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 has answers to Quadratic Equations that can help students prepare for examinations and earn high test scores by covering all of the necessary material. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 are an excellent resource that may help students not only complete the entire curriculum, but also analyse the subjects in-depth.

Quadratic Equations are a part of Algebra. It is derived from a Square. It is a Calculator of Algebraic origin or the second degree. A Quadratic Equation is employed in a variety of situations. A Quadratic equation has several applications in Physics, Engineering, Astronomy, and other fields. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 which are accessible on the Extramarks website provide detailed information of Quadratic Equations for students.

Quadratic equations are second-degree equations in x with a maximum of two solutions. These two answers for x are also known as the Quadratic Equations’ roots and are denoted as (α, β).

In its usual form, the quadratic equation is ax2 + bx + c = 0, where a and b are the coefficients, x is the variable, and c is the constant factor. The first requirement for an equation to be a quadratic equation is that the coefficient of x2 is not zero (a 0). When expressing a quadratic equation in standard form, the x2 term comes first, then the x term, and lastly the constant term. The numeric values a, b, and c are commonly expressed as integral values rather than fractions or decimals.

Quadratic Equations have been described and explained in the NCERT Solutions for Class 10 Maths Chapter 4 Exercise 4.1. Examples are also given along with problems that are present in the NCERT Solutions for Class 10 Maths Chapter 4 Exercise 4.1.

Quadratic Equations are used in many different sectors and daily activities in real life. Some of the professions that involve quadratic equations include astrology, engineering, agriculture, the sciences, the military, and sports. What use do quadratic equations serve, then? In many real-world applications, such as calculating enclosed space areas, object speed, product profit and loss, or bending a piece of equipment for design, Quadratic Equations are employed.

Students are able to use the Quadratic Equations once they understand the principles behind them. The NCERT Solutions for Class 10 Maths Chapter 4 Exercise 4.1 have detailed explanations along with samples. The NCERT Solutions for Class 10 Maths Chapter 4 Exercise 4.1 also provide examples and problems that, when attempted by students, clarify their doubts, if any.

One such real-world illustration is the projection of an item; in this case, Quadratic Equations may be used to calculate the location at which the object will touch the Earth, the object’s journey distance, and the amount of time it will take to reach its peak height.

The NCERT Solutions for Class 10 Maths Chapter 4 Exercise 4.1 are helpful to students as they can solve the problems provided there, which enables them to assess their own progress in understanding the topic.

The Quadratic Equation in Physics are used to analyse motion. So it is a useful tool in Rocket Science. There are many other studies, for example Agriculture, Sports, Construction and Real estate. Military. Engineering. Wireless communications, etc.

Students are advised to visit the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 which are accessible on the Extramarks website to obtain complete assistance. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 can be accessed whenever students require clarification of their doubts or as a reference. .

## NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations

A quadratic equation can be solved using the principles of algebra, factoring strategies where applicable, and the Principle of Zero Products.

The values of x that satisfy the equation are called Solutions of the Equation, and roots or zero of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is only one solution, one says that it is a double root. If all the coefficients are real numbers, there are either two real solutions, or a single real double root, or two complex solutions that are complex conjugates of each other. A Quadratic Equation always has two roots, if complex roots are included; and a double root is counted as two. A Quadratic Equation can be factored into an equivalent equation.

The Details of Quadratic Equations are provided in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 which can be accessed on the Extramarks website. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 can be accessed by students to thoroughly understand the topic, as they are always available.

Topics Covered

The topics which have been covered in the NCERT Solutions Class 10 are Quadratic Equation Zeros and Roots, Quadratic Equation Solvation Using Factorization, Completing the Square Method to Solve a Quadratic Equation, Formula for a Quadratic and Origin of Roots.  The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 provide a list of topics that have been covered in  Class 10th Maths Chapter 4 exercise 4.1. The topics have also been discussed in detail in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 for students. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 are available on the Extramarks website.

### H3 – What is a Quadratic Equation?

An algebraic equation of the second degree in x is a Quadratic Equation. The Quadratic Equation is written as ax2 + bx + c = 0, where x is the variable, a and b are the Coefficients, and c is the Constant Term. First, there must be a term other than Zero in the Coefficient of x2 (a ≠ 0) for an Equation to be a Quadratic Equation. The x2 term is written first when constructing a Quadratic Equation in standard form, then the x term, and finally the Constant term. The numerical values of a, b, and c are often expressed as integral values rather than fractions or decimals.

The complete explanation of Quadratic Equation is available on the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 which are accessible on the Extramarks website. Students can not only benefit, but also practise the Quadratic Equation to be an expert on the topic. The questions and problems available in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 provide the students with an opportunity to solve them and understand the progress of their learning. They can always revisit the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 and clarify doubts and misunderstandings whenever required.

Numerous applications of the quadratic equation may be found in daily life. In order to determine commercial profit, quadratic equations are frequently employed. Businesses must solve a quadratic equation even when dealing with little items to calculate how many of them will be profitable.

In order to understand the utility of the Quadratic Equation, students need to master the topic. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 are always available to the students to obtain assistance regarding the topic.

The Military or Law Enforcement Frequently uses Quadratic Equations to calculate the speed of moving items like Vehicles and Planes. They can also be used by the Military to gauge how far an opponent is from them. Additionally, to forecast where Tanks or Artillery will fall, the Military use Quadratic Equations. It is used by the Police to determine the Gunshot Trajectories. It helps the Traffic Police determine the speeds of the Vehicles involved in Road Accidents.

There is a lot of scope for the students to use the Quadratic Equation in real life. However, to learn and understand the topic properly, students need to master it. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 are always present to assist the students to gather the knowledge. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 prepare students to progress in a seamless manner and be ready for their examinations. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 benefits the students by providing them with examples and questionnaires that help them practise Quadratic Equations.

Quadratic Equations are frequently used in sports. It is now incredibly helpful for both Gaming and Analysis. For instance, a Football Analyst constantly uses Mathematics to ascertain a Team’s or an Athlete’s form. This Analysis has one or two Quadratic Equation elements. Basketball players score by aiming for the exact distance and duration of the throw into the net. A velocity Quadratic Equation can be used to determine the ball’s height. When scoring, players always solve the Equation, but they complete the calculation in a matter of milliseconds in their heads.

The topic is available in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 which are accessible on the Extramarks website.

Construction workers always employ Quadratic Equations to calculate the area before starting a project. People also figure out the sizes of other items like boxes and plots of land. However, building is a fantastic field to use as an example. For instance, the majority of structures are square or rectangular in shape. When constructing a rectangle, one side must cover twice as much space as the other sides. A Quadratic Equation will be created by calculating the area of the materials required to cover that area.

The Extramarks website has the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 which can be used to understand the topic.

A Quadratic Equation may be created by determining an object’s speed. These Equations, for instance, are used by Kayakers to calculate how much speed to apply when moving up or down a river.

Utility of the topic can be understood once it is familiar to the students. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 provide the necessary information on the subject.

When constructing a Satellite Dish, several Quadratic Equation components are used. This is due to the fact that it has to be set up at Specific Angles in order to efficiently pick up signals. The signal is captured by the dish and transmitted to a feed horn, which then broadcasts it to a TV or a station. It is quite difficult to set up a dish to receive a signal from two or three Satellites at once without solving a Quadratic Equation. The experience of a Scientist or Engineer may prevent them from knowing. However, the system must be configured with the correct angles in order for it to function. The principle of Quadratic Equation is available in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1.

The Military or Law Enforcement Frequently uses Quadratic Equations to calculate the speed of moving items like Vehicles and Planes. They can also be used by the Military to gauge how far an opponent is from them. Additionally, to forecast where Tanks or Artillery will fall, the Military use Quadratic Equations. It is used by the Police to determine the Gunshot Trajectories. It helps the Traffic Police determine the speeds of the Vehicles involved in Road Accidents. The understanding of Quadratic Equation has been made easy in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1

### H3 – Standard Form of Quadratic Equation

The typical quadratic equation is written as ax2 + bx + c = 0, where ‘a’ is the leading coefficient and is a non-zero real integer. Because ‘quad’ means square,’ this equation is referred to as ‘quadratic.’ A quadratic equation can be expressed in other ways  than the conventional form. Explanation of the Standard form of Quadratic Equation has been provided in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 which can be accessed on the Extramarks website. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 also provide information about the Fraction form of the Quadratic Equation.

### H3 – Quadratic Equation Example

The Quadratic Equation has been explained in detail in the Solving quadratic equations can be tough, but there are numerous ways available depending on the type of quadratic we are attempting to solve. Factoring, utilising square roots, completing the square, and the quadratic formula are the four ways for solving a quadratic equation. on the Extramarks website. Students can download the PDF file which has been provided in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1. Explanations, examples and questions have been provided to help students solve them and understand the concept better. The PDF file of the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 can be visited offline, which makes it easy to access. Students can get the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 on their handheld device and use the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 for learning and referencing as many times as required.

### H2 – Access NCERT Solutions for Maths Chapter 4 – Quadratic Equations

Students can use the NCERT Solutions for Class 10 Maths, Chapter 4, Exercise 4.1, to find detailed explanations and examples to help them understand and practise quadratic equations.The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 can be accessed on the Extramarks website for student assistance.Questions related to the Quadratic Equations too have been provided in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1. Students should practise the calculations until they are clear with their concepts. If they have any doubts, theycan always revisit the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 on the Extramarks website.

### H2 – Ex 4.1 Class 10 – History of Quadratic Equation

Around 300 BC, Euclid established a Geometrical technique that, while later Mathematicians utilised it to solve Quadratic problems, amounted to finding a length that was the root of a Quadratic Equation in modern notation. Euclid had no concept of Equations, Coefficients, or anything else save Geometrical quantities.

Because the Arabs were unaware of the Hindus’ advancements, they had no negative amounts or Acronyms for their unknowns. However, al-Khwarizmi (around 800) classified distinct forms of Quadratics (although only numerical examples of each). The various forms develop as a result of al-lack Khwarizmi’s zeros and negatives.

Hindu Mathematicians extended Babylonian methods to the point that Brahmagupta (598-665 AD) presents an almost contemporary method that accepts negative values. He frequently utilised Acronyms for the unknown, generally the first letter of a colour, and numerous distinct unknowns appeared in a single puzzle.

The Babylonians were the first to solve Quadratic Equations (about 400 BC). This is an oversimplification since the Babylonians had no concept of ‘Equation.’ They did, however, devise an Algorithmic technique to problem resolution that, in our language, would result in a Quadratic Equation. The approach is essentially one of square completion. However, because the normal solution was a length, all Babylonian questions and answers were positive, or, more precisely, unsigned values.

The Extramarks website contains the  NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 which have the topic of Quadratic Equations explained. The  NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 also provide examples and questions for the students to attempt.

### H3 – Class 10 Maths Chapter 4 Exercise 4.1 – Concept of Quadratic Equation

Here are some fundamental ideas for resolving a Quadratic Equation. There are two strategies students may use to solve a Quadratic Problem. Using the Factorisation approach and a Standard formula. Both the strategies are present in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 for students to access. The strategies can be studied in detail with the help of the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1.

Here are some fundamental ideas for resolving a Quadratic Equation. There are two strategies students may use to solve a Quadratic Problem. Using the Factorisation approach and a Standard formula.This approach may be used to rapidly and simply answer any Quadratic problem. If the equation is one of the types. ax² + bx + c = 0, then the solution will be

x = -b ± √(b² -4ac)/2a.

Students can obtain two different types of numbers by using the + and – signs in this manner.No matter whether a Quadratic Equation can be factored or not, the formula can be used for any sort of Quadratic Equations. In order to better comprehend the procedure. The fundamental ideas have been included in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 for students to understand thoroughly.

Students must identify the factors of the given phrases in factorization. In contrast to the procedure before, the equation we are given can only be solved using this method if it can be factored. For instance, 1, 2, 4, and 8 are factors of 8. The same goes for the variables 1, 2, 3, 4, 6, and 12. Therefore, students can only utilise this approach when the phrases they’ve been given can be factored. Typically, this approach is faster than the alternative.

The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 help students  become familiar with the approaches of Quadratic Equations.

### H3 – Ex 4.1 Class 10 – Solving Techniques of Quadratic Equations

Solving quadratic equations can be tough, but there are numerous ways available depending on the type of quadratic we are attempting to solve. Factoring, utilising square roots, completing the square, and the quadratic formula are the four ways for solving a quadratic equation.

The technique is present in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 for students to learn. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 also provide problems for students to practise the techniques and familiarise themselves.

### H3 – Class 10 Chapter 4 Exercise 4.1: Conversion of Non-Quadratic to Quadratic Equation

This novel approach may be the simplest and fastest way to solve factorable quadratic equations. Its advantages are as follows: easy, quick, methodical, no guessing, no factoring by grouping, and no computing binomials. It employs three aspects in its problem-solving process:

The Rule of Signs for Real Roots of a Quadratic Equation is used to find a better solution method.

When a = 1, use the Diagonal Sum Method to solve simplified quadratic equations of the form x2 + bx + c = 0. This approach can quickly determine the equation two real roots.

The Extramarks website has the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 which provide all the necessary explanations of Quadratic Equation. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 provide problems for students to solve. As students become experts with the Quadratic Equations, it is very easy for them to convert Non-Quadratic Equations to Quadratic Equations and solve problems easily and quickly. It is advisable for students to visit the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 and practise the problems provided there to become familiarwith the principles and method.

### H3 – Class 10 Maths Ex 4.1 – Sum and Product of Roots of a Quadratic Equation

For any quadratic equation, ax2 + bx + c = 0, the sum of the roots, + β = -b/a, and the product of the roots, α × β = c/a. The translation of a quadratic equation in conventional form ax^2 + bx + c = 0 into the simplified form, with a = 1, to facilitate solution.

Practising the Quadratic Equations provides confidence to students to prepare for examinations. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 provide explanations along with questions and problems regarding the topic. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 help students  understand their progress about the topic.

### H3 – Class 10 Maths Ch 4 Ex 4.1- Problems based on Quadratic Equation

The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 have problems which are based on Quadratic Equations which have been explained and examples provided. Students are advised to visit the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 and access the problems provided. Practising the problems will certainly help the students  understand the concept better.

In the NCERT Solutions for Class 10 Maths, Chapter 4, Exercise 4.1, there are problems about finding the area of a field using the Standard Quadratic Equation and finding age where some facts have been provided, as well as problems about speedThe NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 are accessible on the Extramarks website, which is a wonderful tool for the students to use.

### H3 – Why Choose Extramarks?

The Extramarks website has been designed to be a complete solution for students. It has been prepared keeping in mind students of CBSE and SSE.  The information provided in the Extramarks website has been selectively introduced to students according to their development. There is a gradual progression provided to students from Class 1 to Class 12. The NCERT Solutions Class 1, NCERT Solutions Class 2, NCERT Solutions Class 3, NCERT Solutions Class 4 and NCERT Solutions Class 5 have been created for junior school students where they develop into students who are able to read and write along with understanding some concepts. The NCERT Solutions Class 6, NCERT Solutions Class 7 and NCERT Solutions Class 8 are for the middle school students, and the NCERT Solutions Class 9, NCERT Solutions Class 10, NCERT Solutions Class 11 and the NCERT Solutions Class 12 are for students of the senior school. The Extramarks website not only provides information to the students but also accommodates online classes along with explanations, examples, and questionnaires to assist students to develop themselves. The Extramarks website also provides facilities to assess the progress of students in their learning.  The Extramarks website is a wonderful tool which can be used to assist students to fetch better marks in their examinations. As mentioned above, the Extramarks website is always present to assist students from the start to the end of their school lives..

### H3 – NCERT Solutions for Class 10 Maths Chapter 4 Exercises

NCERT Solutions for Class 10 Maths Quadratic Equation The fourth chapter is about comprehending Quadratic Equations and the various methods for calculating their roots. Students preparing for Class 10 Math Chapter 4 will be able to clear all of their concepts on Linear Equations at the root level.It goes through all the main concepts in depth, helping pupils better comprehend the principles.

The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 cover the exercises that need to be practised by students. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 also provide the table of exercise and the number of questions asked to prepare students for examinations. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 provide solutions of the questions to enable students to cross check their answers and find misunderstandings and doubts. Students can visit the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 whenever required and rectify their doubts and misunderstandings. The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 help students completely and preparethem to score high marks in examinations.

The Quadratic Equation definition, standard form of a Quadratic Equation, nature of Roots, idea of Discriminants, Quadratic formula, Factorization technique of solving a Quadratic Equation, and completing the Square method are all covered in NCERT Solutions for Class 10 Maths Chapter 4.

The fourth chapter of NCERT Solutions for Class 10 Maths has four tasks. There are 24 problems in Class 10 Maths Chapter 4 Quadratic Equations, 15 of which are basic, 5 of which are intermediate, and 4 of which are difficult. These questions are addressed one by one. By completing these assignments, students will be able to answer all quadratic equation-based problems. Furthermore, the questions are addressed in several ways to assist students in mastering basicquadratic equation concepts.

The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 which are available on the Extramarks have examples that explain the concept of Quadratic Equations for students to understand.

Q.1

$\begin{array}{l}\text{Check whether the following are quadratic equations :}\\ \left(\text{i}\right)\text{}{\left(\mathrm{x}+1\right)}^{2}=2\left(\mathrm{x}-3\right)\text{}\left(\text{ii}\right)\text{}{\mathrm{x}}^{2}-2\mathrm{x}=-2\left(3-\mathrm{x}\right)\\ \left(\mathrm{iii}\right)\text{\hspace{0.17em}\hspace{0.17em}}\left(\mathrm{x}-2\right)\left(\mathrm{x}+1\right)=\left(\mathrm{x}-1\right)\left(\mathrm{x}+3\right)\text{}\left(\text{iv}\right)\text{}\left(\mathrm{x}-3\right)\left(2\mathrm{x}+1\right)=\mathrm{x}\left(\mathrm{x}+5\right)\\ \left(\mathrm{v}\right)\text{}\left(2\mathrm{x}-1\right)\left(\mathrm{x}-3\right)=\left(\mathrm{x}+5\right)\left(\mathrm{x}-1\right)\text{}\left(\text{vi}\right){\text{x}}^{2}+3\mathrm{x}+1={\left(\mathrm{x}-2\right)}^{2}\\ \left(\mathrm{vii}\right)\text{\hspace{0.17em}}{\left(\mathrm{x}+2\right)}^{3}=2\mathrm{x}\left({\mathrm{x}}^{2}-1\right)\text{}\left(\text{viii}\right){\text{x}}^{3}-4{\mathrm{x}}^{2}-\mathrm{x}+1={\left(\mathrm{x}-2\right)}^{3}\end{array}$

Ans.

$\begin{array}{l}\text{(i)}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\left(\mathrm{x}+1\right)}^{2}=2\left(\mathrm{x}-3\right)\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}+2\mathrm{x}+1=2\mathrm{x}-6\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}+2\mathrm{x}+1-2\mathrm{x}+6=0\\ ⇒{\mathrm{x}}^{2}+7=0\\ \text{It is of the form a}{\mathrm{x}}^{2}+\mathrm{bx}+\mathrm{c}=0.\\ \text{Therefore, the given equation is a quadratic equation.}\\ \text{(ii)}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}-2\mathrm{x}=\left(-2\right)\left(3-\mathrm{x}\right)\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}-2\mathrm{x}=2\mathrm{x}-6\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}-2\mathrm{x}-2\mathrm{x}+6=0\\ ⇒{\mathrm{x}}^{2}-4\mathrm{x}+6=0\\ \text{It is of the form a}{\mathrm{x}}^{2}+\mathrm{bx}+\mathrm{c}=0.\\ \text{Therefore, the given equation is a quadratic equation.}\\ \text{(iii)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\mathrm{x}-2\right)\left(\mathrm{x}+1\right)=\left(\mathrm{x}-1\right)\left(\mathrm{x}+3\right)\\ ⇒\text{}{\mathrm{x}}^{2}-\mathrm{x}-2={\mathrm{x}}^{2}+2\mathrm{x}-3\\ ⇒\text{}{\mathrm{x}}^{2}-\mathrm{x}-2-{\mathrm{x}}^{2}-2\mathrm{x}+3=0\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}-3\mathrm{x}+1=0\\ \text{It is not of the form}{\mathrm{ax}}^{2}+\mathrm{bx}+\mathrm{c}=0.\\ \text{Therefore, the given equation is not a quadratic equation.}\\ \text{(iv)}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\mathrm{x}-3\right)\left(2\mathrm{x}+1\right)=\mathrm{x}\left(\mathrm{x}+5\right)\\ ⇒\text{2}{\mathrm{x}}^{2}-5\mathrm{x}-3={\mathrm{x}}^{2}+5\mathrm{x}\\ ⇒\text{2}{\mathrm{x}}^{2}-5\mathrm{x}-3-{\mathrm{x}}^{2}-5\mathrm{x}=0\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}-10\mathrm{x}-3=0\\ \text{It is of the form}{\mathrm{ax}}^{2}+\mathrm{bx}+\mathrm{c}=0.\\ \text{Therefore, the given equation is a quadratic equation.}\\ \text{(v)}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(2\mathrm{x}-1\right)\left(\mathrm{x}-3\right)=\left(\mathrm{x}+5\right)\left(\mathrm{x}-1\right)\\ ⇒\text{2}{\mathrm{x}}^{2}-7\mathrm{x}+3={\mathrm{x}}^{2}+4\mathrm{x}-5\\ ⇒\text{2}{\mathrm{x}}^{2}-7\mathrm{x}+3-{\mathrm{x}}^{2}-4\mathrm{x}+5=0\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}-11\mathrm{x}+8=0\\ \text{It is of the form}{\mathrm{ax}}^{2}+\mathrm{bx}+\mathrm{c}=0.\\ \text{Therefore, the given equation is a quadratic equation.}\\ \text{(vi)}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}+3\mathrm{x}+1={\left(\mathrm{x}-2\right)}^{2}\\ ⇒{\mathrm{x}}^{2}+3\mathrm{x}+1={\mathrm{x}}^{2}-4\mathrm{x}+4\\ ⇒{\mathrm{x}}^{2}+3\mathrm{x}+1-{\mathrm{x}}^{2}+4\mathrm{x}-4=0\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}}7\mathrm{x}-3=0\\ \text{It is not of the form}{\mathrm{ax}}^{2}+\mathrm{bx}+\mathrm{c}=0.\\ \text{Therefore, the given equation is not a quadratic equation.}\\ \text{(vii)}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\left(\mathrm{x}+2\right)}^{3}=2\mathrm{x}\left({\mathrm{x}}^{2}-1\right)\\ ⇒{\mathrm{x}}^{3}+6{\mathrm{x}}^{2}+12\mathrm{x}+8=2{\mathrm{x}}^{3}-2\mathrm{x}\\ ⇒{\mathrm{x}}^{3}+6{\mathrm{x}}^{2}+12\mathrm{x}+8-2{\mathrm{x}}^{3}+2\mathrm{x}=0\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}}-{\mathrm{x}}^{3}+6{\mathrm{x}}^{2}+14\mathrm{x}+8=0\\ \text{It is not of the form}{\mathrm{ax}}^{2}+\mathrm{bx}+\mathrm{c}=0.\\ \text{Therefore, the given equation is not a quadratic equation.}\\ \text{(viii)}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{3}-4{\mathrm{x}}^{2}-\mathrm{x}+1={\left(\mathrm{x}-2\right)}^{3}\\ ⇒\text{\hspace{0.17em}}{\mathrm{x}}^{3}-4{\mathrm{x}}^{2}-\mathrm{x}+1={\mathrm{x}}^{3}-6{\mathrm{x}}^{2}+12\mathrm{x}-8\\ ⇒{\mathrm{x}}^{3}-4{\mathrm{x}}^{2}-\mathrm{x}+1-{\mathrm{x}}^{3}+6{\mathrm{x}}^{2}-12\mathrm{x}+8=0\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}}2{\mathrm{x}}^{2}-13\mathrm{x}+9=0\\ \text{It is of the form}{\mathrm{ax}}^{2}+\mathrm{bx}+\mathrm{c}=0.\\ \text{Therefore, the given equation is a quadratic equation.}\end{array}$

Q.2 Represent the following situations in the form of quadratic equations:

1. 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.

2. The product of two consecutive positive integers is 306. We need to find the integers.

3. 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.

4. 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.

Ans.

$\begin{array}{l}\text{(i) Let the breadth of the plot be}\mathrm{x}\text{}\mathrm{m}\text{. Then, as per given}\\ \text{information, the length of the plot is}\left(2\mathrm{x}+1\right)\text{}\mathrm{m}\text{.}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}Also, given that area of the rectangular plot is}528\text{}{\mathrm{m}}^{2}.\\ \text{Therefore,}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(2\mathrm{x}+1\right)\mathrm{x}=528\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left[\text{Area of a rectangle}=\text{Length}×\text{Breadth}\right]\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}or\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}2{\mathrm{x}}^{2}+\mathrm{x}-528=0.\\ \text{(ii) Let two consecutive positive integers are}\mathrm{x}\text{and}\mathrm{x}+1.\\ \text{Then we have,}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\mathrm{x}\left(\mathrm{x}+1\right)=306\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}+\mathrm{x}-306=0\\ \text{(iii) Let Rohan’s present age is}\mathrm{x}\text{years. Then Rohan’s mother}\\ \text{is}\mathrm{x}+26\text{years old.}\\ \text{According to question,}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\mathrm{x}+3\right)\left(\mathrm{x}+26+3\right)=360\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(\mathrm{x}+3\right)\left(\mathrm{x}+29\right)=360\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}+32\mathrm{x}+87-360=0\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}+32\mathrm{x}-273=0\\ \text{(iv) Let uniform speed of the train is}\mathrm{x}\text{km/h. Then the time}\\ \text{\hspace{0.17em} taken to travel the distance of 480 km is}\frac{480}{\mathrm{x}}\text{hours.}\\ \text{Again, when the speed is (}\mathrm{x}-8\right)\text{km/h then the time taken}\\ \text{to travel the distance of 480 km is}\frac{480}{\mathrm{x}-8}\text{hours.}\\ \text{According to question,}\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\frac{480}{\mathrm{x}-8}-\frac{480}{\mathrm{x}}=3\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}480\left(\mathrm{x}-\mathrm{x}+8\right)=3\mathrm{x}\left(\mathrm{x}-8\right)\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\frac{480×8}{3}={\mathrm{x}}^{2}-8\mathrm{x}\\ ⇒\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}{\mathrm{x}}^{2}-8\mathrm{x}-1280=0\end{array}$

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### 1. What are the main topics covered in NCERT Solutions Class 10 Maths Chapter 4?

The cornerstone of Chapter 4 of Class 10 Maths is Quadratic Equations. The key topics covered in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 are how to Mathematically represent the given problem statements, what is the standard form of a Quadratic Equation, and how to solve Quadratic Equations by Factoring and completing the Squares, which is an important topic that requires regular practise.

### 2. How does one solve Quadratic Equations in Class 10?

If anyone wants to learn how to solve Quadratic Equations in Class 10, they may acquire NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 from the Extramarks website. Solutions have been provided in clear terms. Students may readily understand the Equations. To get the Roots, students must apply the Quadratic formula. They are able to calculate the sum and product of both Roots. The technique is simple and well-explained for clarity.

### 3. In 10th Maths Ex. 4.1, what is the standard form of a quadratic equation?

A quadratic equation in x is an equation of the form ax2 + bx + c = 0, where a, b, and c are real values, and a = 0. A quadratic equation is, for example, 2×2 + x – 300 = 0. Similarly, quadratic equations include 2×2 – 3x + 1 = 0, 4x – 3×2 + 2 = 0, and 1 – x2 + 300 = 0. In fact, a quadratic equation is any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2. The standard form of the quadratic equation is obtained by writing the terms of p(x) in descending order of their degrees.

That is, the conventional form of a quadratic equation is ax2 + bx + c = 0.

The NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 contain many more Frequently Asked Questions which helps to clarify doubts of students.

### 4. What are the main themes covered in NCERT Solutions Class 10 Maths Chapter 4?

Maths Chapter 4 for Class 10 NCERT answers the topic of Quadratic Equations and the many methods of determining their Roots are covered in the book Quadratic Equations. A Quadratic Equation is written as ax2 + bx + c = 0, where a, b, and c are real-number values and an is a non-zero number. This is sometimes referred to as the Quadratic Equation’s standard form. It’s important to note that a lot of people think the Babylonians were the first to figure out how to solve Quadratic Equations. For instance, they were able to solve the Equivalent of a Quadratic Equation that is  finding two positive Integers with a specified positive sum and a given positive product.

In addition, the Greek mathematician Euclid created a Geometrical method for determining lengths that, in modern terms, are the solutions to Quadratic Equations. Kids are taught how to solve these Equations via the Factorization approach and the completing the Square method in the NCERT Solutions for Class 10 Maths Chapter 4 exercise 4.1 on Quadratic Equations. Two Coincident Roots will be obtained, if b2 – 4ac = 0, and no roots will be present, if b2 – 4ac < 0, according to the Quadratic formula for determining the roots of the Equation. The use of Quadratic Equations in practical contexts will also be explored by the students.