# NCERT Solutions for Class 10 Maths Chapter 4 Quadratic Equations

Exercise:4.1

Exercise:4.2

Exercise:4.3

Exercise:4.4

## Types of questions asked from class 10 maths chapter 4 Quadratic Equations

• Representation of situation in a quadratic equation.
• Checking of an equation is quadratic or not.
• The solution of the quadratic equation using completing the square method.
• Solving a quadratic equation using the Sridharacharya formula.
• Product of roots in a quadratic equation.
• Sum of roots in a quadratic equation.

## NCERT solutions for class 10 maths chapter 4 Quadratic Equations Excercise: 4.1

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.

Given equation  can be written as:

i.e.,

This equation is of type: .

Hence, the given equation is a quadratic equation.

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.

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.

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.

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.

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.

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.

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

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 .

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 .

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

## Q1 (i) Find the roots of the following quadratic equations by factorization:

Factorization gives,

Hence, the roots of the given quadratic equation are .

Factorisation gives,

Hence, the roots of the given quadratic equation are

Factorization gives,

Hence, the roots of the given quadratic equation are

Solving the quadratic equations, we get

Factorization gives,

Hence, the roots of the given quadratic equation are

Factorization gives,

Hence, the roots of the given quadratic equation are

.

From Example 1 we get:

Equations:

(i)

Solving by factorization method:

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:

Factorization gives,

Hence, the roots of the given quadratic equation are .

Therefore, the number of toys on that day was

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 .

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.

## NCERT solutions for class 10 maths chapter 4 Quadratic Equations Excercise: 4.3

Given equation:

On dividing both sides of the equation by 2, we obtain

Given equation:

On dividing both sides of the equation by 2, we obtain

Adding and subtracting    in the equation, we get

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:

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).

(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.

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

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 '30-m'.

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 30-12 = 18.

Or,

The marks obtained in Mathematics is 13, then marks in English will be 30-13 = 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,

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,

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 .

## NCERT solutions for class 10 maths chapter 4 Quadratic Equations Excercise: 4.4

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.

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.

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,

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 .

## NCERT solutions for class 10 maths chapter wise

 Chapter No. Chapter Name Chapter 1 CBSE NCERT solutions for class 10 maths chapter 1 Real Numbers Chapter 2 NCERT solutions for class 10 maths chapter 2 Polynomials 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 Solutions of NCERT class 10 maths chapter 6 Triangles 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 CBSE NCERT solutions class 10 maths chapter 10 Circles Chapter 11 NCERT solutions  for class 10 maths chapter 11 Constructions 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 NCERT solutions for class 10 maths chapter 14 Statistics Chapter 15 Solutions of NCERT class 10 maths chapter 15 Probability

## How to use NCERT solutions for class 10 maths chapter 4 Quadratic Equations?

• 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!