NCERT solutions for class 10 maths chapter 4 Quadratic Equations: In the previous class you have studied polynomials. A polynomial having degree two is called a quadratic equation. Solutions of NCERT class 10 maths chapter 4 Quadratic Equations can be a good assistant for you while preparing this chapter. When we equate the quadratic polynomial to zero, then we get a quadratic equation. That is an equation of the form ax^{2}+bx+c=0 with 'a' not equal to zero is known as quadratic equations. There are many reallife problems where we can form quadratic equations to solve it. CBSE NCERT solutions for class 10 maths chapter 4 Quadratic Equations introduces the concepts of quadratic equations and different methods to solve it. NCERT solutions for class 10 maths chapter 4 Quadratic Equations is providing an indepth and step by step solution to each question. For example, Ramu has a rectangular plot. He forgot the length and breadth of the plot. But he remembers that area of the plot is 400 m^{2} and length is 9m more than the breadth. What should be the length and breadth of the plot? To solve this question let's consider the length to be x then the breadth will be x9 and the area is (x9)x=400. This is how a word problem is converted to a quadratic equation. NCERT Solutions explains different methods to solve it. Four exercises of this chapter are explained below.
Q1 (i) Check whether the following are quadratic equations :
We have L.H.S.
Therefore, can be written as:
i.e.,
Or
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (ii) Check whether the following are quadratic equations :
Given equation can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (iii) Check whether the following are quadratic equations :
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
The equation is of the type: .
Hence, the given equation is not a quadratic equation since a=0.
Q1 (iv) Check whether the following are quadratic equations :
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (v) Check whether the following are quadratic equations :
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q1 (vi) Check whether the following are quadratic equations :
L.H.S.
and R.H.S can be written as:
i.e.,
This equation is NOT of type: .
Here a=0, hence, the given equation is not a quadratic equation.
Q1 (vii) Check whether the following are quadratic equations :
L.H.S. can be written as:
and R.H.S can be written as:
i.e.,
This equation is NOT of type: .
Hence, the given equation is not a quadratic equation.
Q1 (viii) Check whether the following are quadratic equations :
L.H.S. ,
and R.H.S can be written as:
i.e.,
This equation is of type: .
Hence, the given equation is a quadratic equation.
Q2 (i) Represent the following situations in the form of quadratic equations : The area of a rectangular plot is . The length of the plot (in meters) is one more than twice its breadth. We need to find the length and breadth of the plot.
Given the area of a rectangular plot is .
Let the breadth of the plot be .
Then, the length of the plot will be: .
Therefore the area will be:
which is equal to the given plot area .
Hence, the length and breadth of the plot will satisfy the equation
Q2 (ii) Represent the following situations in the form of quadratic equations : The product of two consecutive positive integers is 306. We need to find the integers.
Given the product of two consecutive integers is
Let two consecutive integers be and .
Then, their product will be:
Or .
Hence, the two consecutive integers will satisfy this quadratic equation .
Q2 (iii) Represent the following situations in the form of quadratic equations: 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.
Let the age of Rohan be years.
Then his mother age will be: years.
After three years,
Rohan's age will be years and his mother age will be years.
Then according to question,
The product of their ages 3 years from now will be:
Or
Hence, the age of Rohan satisfies the quadratic equation .
Q2 (iv) Represent the following situations in the form of quadratic equations : 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.
Let the speed of the train be km/h.
The distance to be covered by the train is .
The time taken will be
If the speed had been less, the time taken would be: .
Now, according to question
Dividing by 3 on both the side
Hence, the speed of the train satisfies the quadratic equation
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are .
Q1 (ii) Find the roots of the following quadratic equations by factorization:
Given the quadratic equation:
Factorisation gives,
Hence, the roots of the given quadratic equation are
Q1 (iii) Find the roots of the following quadratic equations by factorization:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are
Q1 (iv) Find the roots of the following quadratic equations by factorization:
Given the quadratic equation:
Solving the quadratic equations, we get
Factorization gives,
Hence, the roots of the given quadratic equation are
Q1 (v) Find the roots of the following quadratic equations by factorization:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are
.
Q2 Solve the problems given in Example 1. (i) (ii)
From Example 1 we get:
Equations:
(i)
Solving by factorization method:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are .
Therefore, John and Jivanti have 36 and 9 marbles respectively in the beginning.
(ii)
Solving by factorization method:
Given the quadratic equation:
Factorization gives,
Hence, the roots of the given quadratic equation are .
Therefore, the number of toys on that day was
Q3 Find two numbers whose sum is 27 and the product is 182.
Let two numbers be x and y.
Then, their sum will be equal to 27 and the product equals 182.
...............................(1)
.................................(2)
From equation (2) we have:
Then putting the value of y in equation (1), we get
Solving this equation:
Hence, the two required numbers are .
Q4 Find two consecutive positive integers, the sum of whose squares is 365.
Let the two consecutive integers be
Then the sum of the squares is 365.
.
Hence, the two consecutive integers are .
Let the length of the base of the triangle be .
Then, the altitude length will be: .
Given if hypotenuse is .
Applying the Pythagoras theorem; we get
So,
Or
But, the length of the base cannot be negative.
Hence the base length will be .
Therefore, we have
Altitude length and Base length
Let the number of articles produced in a day
The cost of production of each article will be
Given the total production on that day was .
Hence we have the equation;
But, x cannot be negative as it is the number of articles.
Therefore, and the cost of each article
Hence, the number of articles is 6 and the cost of each article is Rs.15.
Q1 (i) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Given equation:
On dividing both sides of the equation by 2, we obtain
Q1 (ii) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Given equation:
On dividing both sides of the equation by 2, we obtain
Adding and subtracting in the equation, we get
Q1 (iii) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Given equation:
On dividing both sides of the equation by 4, we obtain
Adding and subtracting in the equation, we get
Hence there are the same roots and equal:
Q2 (iv) Find the roots of the following quadratic equations, if they exist, by the method of completing the square
Given equation:
On dividing both sides of the equation by 2, we obtain
Adding and subtracting in the equation, we get
Here the real roots do not exist (in the higher studies we will study how to find the root of such equations).
Q2 Find the roots of the quadratic equations given in Q.1 above by applying the quadratic formula.
(i)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Therefore, the real roots are:
(ii)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Therefore, the real roots are:
(iii)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Therefore, the real roots are:
(iv)
The general form of a quadratic equation is : , where a, b, and c are arbitrary constants.
Hence on comparing the given equation with the general form, we get
And the quadratic formula for finding the roots is:
Substituting the values in the quadratic formula, we obtain
Here the term inside the root is negative
Therefore there are no real roots for the given equation.
Q3 (i) Find the roots of the following equations:
Given equation:
So, simplifying it,
Comparing with the general form of the quadratic equation: , we get
Now, applying the quadratic formula to find the roots:
Therefore, the roots are
Q3 (ii) Find the roots of the following equations:
Given equation:
So, simplifying it,
or
Can be written as:
Hence the roots of the given equation are:
Let the present age of Rehman be years.
Then, 3 years ago, his age was years.
and 5 years later, his age will be years.
Then according to the question we have,
Simplifying it to get the quadratic equation:
Hence the roots are:
However, age cannot be negative
Therefore, Rehman is 7 years old in the present.
Let the marks obtained in Mathematics be 'm' then, the marks obtain in English will be '30m'.
Then according to the question:
Simplifying to get the quadratic equation:
Solving by the factorizing method:
We have two situations when,
The marks obtained in Mathematics is 12, then marks in English will be 3012 = 18.
Or,
The marks obtained in Mathematics is 13, then marks in English will be 3013 = 17.
Let the shorter side of the rectangle be x m.
Then, the larger side of the rectangle wil be .
Diagonal of the rectangle:
It is given that the diagonal of the rectangle is 60m more than the shorter side.
Therefore,
Solving by the factorizing method:
Hence, the roots are:
But the side cannot be negative.
Hence the length of the shorter side will be: 90 m
and the length of the larger side will be
Given the difference of squares of two numbers is 180.
Let the larger number be 'x' and the smaller number be 'y'.
Then, according to the question:
and
On solving these two equations:
Solving by the factorizing method:
As the negative value of x is not satisfied in the equation:
Hence, the larger number will be 18 and a smaller number can be found by,
putting x = 18, we obtain
.
Therefore, the numbers are or .
Let the speed of the train be
Then, time taken to cover will be:
According to the question,
Making it a quadratic equation.
Now, solving by the factorizing method:
However, the speed cannot be negative hence,
The speed of the train is .
Let the time taken by the smaller pipe to fill the tank be
Then, the time taken by the larger pipe will be: .
The fraction of the tank filled by a smaller pipe in 1 hour:
The fraction of the tank filled by the larger pipe in 1 hour.
Given that two water taps together can fill a tank in hours.
Therefore,
Making it a quadratic equation:
Hence the roots are
As time is taken cannot be negative:
Therefore, time is taken individually by the smaller pipe and the larger pipe will be and hours respectively.
Let the average speed of the passenger train be .
Given the average speed of the express train
also given that the time taken by the express train to cover 132 km is 1 hour less than the passenger train to cover the same distance.
Therefore,
Can be written as quadratic form:
Roots are:
As the speed cannot be negative.
Therefore, the speed of the passenger train will be and
The speed of the express train will be .
Let the sides of the squares be . (NOTE: length are in meters)
And the perimeters will be: respectively.
Areas respectively.
It is given that,
.................................(1)
.................................(2)
Solving both equations:
or putting in equation (1), we obtain
Solving by the factorizing method:
Here the roots are:
As the sides of a square cannot be negative.
Therefore, the sides of the squares are and .
For a quadratic equation, the value of discriminant determines the nature of roots and is equal to:
If D>0 then roots are distinct and real.
If D<0 then no real roots.
If D= 0 then there exists two equal real roots.
Given the quadratic equation, .
Comparing with general to get the values of a,b,c.
Finding the discriminant:
Here D is negative hence there are no real roots possible for the given equation.
Q1 (ii) Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
Here the value of discriminant =0, which implies that roots exist and the roots are equal.
The roots are given by the formula
So the roots are
The value of the discriminant
The discriminant > 0. Therefore the given quadratic equation has two distinct real root
roots are
So the roots are
For two equal roots for the quadratic equation:
The value of the discriminant .
Given equation:
Comparing and getting the values of a,b, and, c.
The value of
Or,
Q2 (ii) Find the values of k for each of the following quadratic equations so that they have two equal roots
For two equal roots for the quadratic equation:
The value of the discriminant .
Given equation:
Can be written as:
Comparing and getting the values of a,b, and, c.
The value of
But is NOT possible because it will not satisfy the given equation.
Hence the only value of is 6 to get two equal roots.
Let the breadth of mango grove be .
Then the length of mango grove will be .
And the area will be:
Which will be equal to according to question.
Comparing to get the values of .
Finding the discriminant value:
Here,
Therefore, the equation will have real roots.
And hence finding the dimensions:
As negative value is not possible, hence the value of breadth of mango grove will be 20m.
And the length of mango grove will be:
Let the age of one friend be
and the age of another friend will be:
4 years ago, their ages were, and .
According to the question, the product of their ages in years was 48.
or
Now, comparing to get the values of .
Discriminant value
As .
Therefore, there are no real roots possible for this given equation and hence,
This situation is NOT possible.
Let us assume the length and breadth of the park be respectively.
Then, the perimeter will be
The area of the park is:
Given :
Comparing to get the values of a, b and c.
The value of the discriminant
As .
Therefore, this equation will have two equal roots.
And hence the roots will be:
Therefore, the length of the park,
and breadth of the park .
Chapter No. 
Chapter Name 
Chapter 1 
CBSE NCERT solutions for class 10 maths chapter 1 Real Numbers 
Chapter 2 

Chapter 3 
Solutions of NCERT class 10 maths chapter 3 Pair of Linear Equations in Two Variables 
Chapter 4 
CBSE NCERT solutions for class 10 maths chapter 4 Quadratic Equations 
Chapter 5 
NCERT solutions for class 10 chapter 5 Arithmetic Progressions 
Chapter 6 

Chapter 7 
CBSE NCERT solutions for class 10 maths chapter 7 Coordinate Geometry 
Chapter 8 
NCERT solutions for class 10 maths chapter 8 Introduction to Trigonometry 
Chapter 9 
Solutions of NCERT class 10 maths chapter 9 Some Applications of Trigonometry 
Chapter 10 

Chapter 11 

Chapter 12 
Solutions of NCERT class 10 chapter maths chapter 12 Areas Related to Circles 
Chapter 13 
CBSE NCERT solutions class 10 maths chapter 13 Surface Areas and Volumes 
Chapter 14 

Chapter 15 
First of all list down all the questions in which you need assistance and go through the NCERT solution of that particular question.
When you complete the first step then your next target should be previous papers. You can pick past year papers and practice them thoroughly.
Once you complete NCERTs and previous year papers, try to solve the questions of that particular chapter from different state board books.
Keep working hard & happy learning!
Q1. Check whether the following are quadratic equations :
(viii)
Q1. Find the roots of the following quadratic equations, if they exist, by the method of completing the square:
(i)
Q10. An express train takes 1 hour less than a passenger train to travel 132 km between Mysore and Bangalore (without taking into consideration the time they stop at intermediate stations). If the average speed of the express train is 11km/h more than that of the passenger train, find the average speed of the two trains.