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

Quadratic equation Class 10 Maths NCERT Solutions Chapter 4 is concerned with understanding quadratic equations and numerous methods for determining their roots. Students preparing for class 10 maths chapter 4 will be able to clear all their concepts on linear equations at the root level. Expert faculty of Toppr produced these solutions to assist students with their first term exam preparations. It covers all major concepts in detail, allowing students to understand the ideas better.

NCERT Solutions for class 10 maths chapter 4 of the section, concentrates on the essential concepts such as the quadratic equation definition, standard form of a quadratic equation, nature of roots, concept of discriminants, quadratic formula, factorization technique of solving a quadratic equation, and completing the square method. All of these solutions are designed with the new CBSE pattern in mind, so that students have a complete understanding of their tests.

Quadratic equation class 10 solutions are very useful for getting good grades in tests and properly preparing you with all of the important concepts. These NCERT Solutions are valuable tools that can assist you not only in covering the full syllabus but also in providing an in-depth analysis of the subjects. The Class 10 Maths NCERT Solutions Chapter 4 are available in pdf format below, and some of them are also included in the exercises.

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

Exercise 4.1
Question 1
Check whether the following are quadratic equations :
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)
(viii)
Medium
Solution Verified by Toppr
i)

ii)

iii)

iv)

v)

vi)

vii)

viii)
Question 2
Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is . The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

(ii) The product of two consecutive positive integers is 306. We need to find the integers.

(iii) Rohans 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 Rohans present age.

(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
Hard
Solution Verified by Toppr
i)
Let the breadth be m and the length will be m.
Area
Area

ii)
Let one number be then the next number will be
The numbers are and

iii)
Let Rohan's present age yrs.
Then his mother's present age yrs
After yrs
Rohan's age yrs
His mother's age yrs
So, Rohan's present age yrs.

iv)
Let the speed of the train km/hr
The speed of the train is km/hr.
Exercise 4.2
Question 1
Find the roots of the following quadratic equations by factorisation:
(i)
(ii)
(iii)
(iv)
(v)
Medium
Solution Verified by Toppr
i)

ii)

iii)

iv)

v)
Question 2
Solve:

(i) John and Jivanti together have marbles. Both of them lost marbles each, and the product of the number of marbles they now have is . We would like to find out how many marbles they had to start with.

(ii) 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 minus the number of toys produced in a day. On a particular day, the total cost of production was . We would like to find out the number of toys produced on that day.
Hard
Solution Verified by Toppr
Let John have marbles, so Jayanti will have marbles.
After losing marbles,
John will have marbles
Jivanti will have marbles
As per the given condition,
or
or

So, if John has marbles, Jivanti will have marbles.
If John has marbles, Jivanti will have marbles.

Let the total number of toys produced in a day be .
Cost of production of one toy
or
or

So, total number of toys produced in a day are or .
Question 3
Find two numbers whose sum is 27 and product is 182.
Medium
Solution Verified by Toppr
Let the first number be then the second number will be

$x(27-x)=182$\$

If the first number is then the second number is and if the first number is then the second number is .
Question 4
Two consecutive positive integers, sum of whose squares is are
A
B
C
D
Medium
Solution Verified by Toppr
Let the two consecutive positive integers be and
Then,

Using the quadratic formula, we get

and
But is given to be a positive integer.
Hence, the two consecutive positive integers are and .
Question 5
The altitude of a right triangle is less then its base. If the hypotenuse is , find the other two sides.
Hard
Solution Verified by Toppr

Let be the base of the triangle, then the altitude will be .

By Pythagoras theorem,

Since the side of the triangle cannot be negative, so the base of the triangle is and the altitude of the triangle will be .

Question 6
A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was more than twice the number of articles produced on that day. If the total cost of production on that day was Rs find the number of articles produced and the cost of each article.
Medium
Solution Verified by Toppr
Let there be articles.

Then the production cost of each article

Total production cost

Number cannot be negative.

So, number of articles and the cost of each article
Exercise 4.3
Question 1
Find the roots of the following quadratic equations, if they exist, by the method of completing the square :
(i)                    (ii)
(iii)             (iv)
Medium
Solution Verified by Toppr

Taking square root on both sides

Taking square root on both sides

same roots.

Hence, solved.

Question 2
Find the roots of the quadratic equation by applying the quadratic formula
A
B

C

D
None of these
Easy
Solution Verified by Toppr
Given equation is
Hence,
Therefore,

Thus, and
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### NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations – Brief Overview

A quadratic equation in the variable x is an equation of the form ax2 + bx + c = 0, where a, b, c are real numbers, a ≠ 0. In fact, any equation of the form p(x) = 0, where p(x) is a polynomial of degree 2, having a single variable, is a quadratic equation.

ax2 + bx + c = 0, a ≠ 0 is called the standard form of a quadratic equation.

For example, 2x2 + x – 30 = 0, 4x2 -2x + 5 = 0, 3x – 4x2 + 2 = 0 are all quadratic equations.

#### 2. Zeroes/Roots of a Quadratic Equation

A real number α is called a root of the quadratic equation ax2 + bx + c = 0,  a ≠ 0 if aα2 + bα + c = 0. We also say that x = α is a solution of the quadratic equation, or that α satisfies the quadratic equation. Note that the zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same.

A quadratic equation can only have two roots/zeroes.

#### 3. Solution of a Quadratic Equation by Factorisation Method

We obtain the roots of a quadratic equation, ax2 + bx + c = 0 using this method by factoring the LHS into two linear parts and equating each element to zero. For example,

2x2 – 5x + 3 = 0

We split the middle term,

2x2 – 2x – 3x + 3 = 0   …..(i)

2x (x – 1) –3(x – 1) = 0

(2x – 3)(x – 1) = 0

2x – 3 = 0 or x – 1 = 0

x = 3/2 or x = 1

Necessary Condition - The product of the first and last terms of eq. (i) should be equal to the product of the second and third terms of the same equation.

#### 4. Solution of a Quadratic Equation by Completing the Square Method

This method involves adding and removing the appropriate constant terms to transform the L.H.S. of a quadratic equation that is not a perfect square into the sum or difference of a perfect square and a constant.

The roots of a quadratic equation ax2 + bx + c = 0 are given by: provided b2 – 4ac ≥ 0

#### 6. Nature of Roots

The value of (b2 – 4ac) is known as the discriminant of the equation and is denoted as D.

1. If D = 0, then the two roots are real and equal
2. If D > 0, then the two roots are real and unequal
3. If D < 0, then the two roots are not real
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### Frequently Asked Questions on NCERT Class 10 Maths Chapter 4 : Quadratic Equations

Q1. Why Should I Practice NCERT Solutions Class 10 Maths Chapter 4?

Answer: Quadratic equation is a topic that is not only essential in mathematics but also plays a vital part in many real-life events. The quadratic equation can be used to calculate the length and width of a garden. You can plan the quantity of grass carpet needed for the garden based on this information. Quadratic equations are frequently employed in astronomy, science, and architecture. Because of its wide range of applications, students should thoroughly practise the NCERT Solutions Class 10 Maths Chapter 4.

Q2. How many exercises are there in Chapter 4 of Class 10 Maths?

Answer: There are four exercises in the fourth chapter of NCERT Solutions for Class 10 Maths. Class 10 Maths Chapter 4 Quadratic Equations contains a total of 24 questions, 15 of which are simple, 5 of which are intermediate, and 4 of which are challenging. These questions are answered step by step. Students can answer all quadratic equation-based questions by completing these activities. In addition, the problems are answered in more than one way to help students learn basic quadratic equation ideas.

Q3. What major topics are addressed in NCERT Solutions Class 10 Maths Chapter 4?

Answer: Quadratic Equations are the foundation of Chapter 4 of Class 10 Maths. The important topics covered in NCERT Solutions Class 10 Maths Chapter 4 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 essential topic that requires regular practice.

Q4. In Class 10, how do you solve Quadratic Equations?

Answer: If you want to learn how to solve quadratic equations in Class 10, you can use the Toppr website or app to access NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations. All solutions are written in simple language by specialists. The equations are easily understood by students. Students must use the quadratic formula to discover the roots. They can compute the sum and product of both roots. The procedure is straightforward and well-explained for clarity.

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