Chpater 4. Quadratic Equations — Class 10 Mathematics NCERT Solutions | CBSE
Chpater 4. Quadratic Equations – NCERT Class 10 Mathematics Solutions | CBSE
Chpater 4. Quadratic Equations
Chapter 4. Quadratic Equations |
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Exercise 4.1 complete solution Exercise 4.2 complete solution Exercise 4.3 complete solution Exercise 4.4 complete solution |
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Important notes : 1. A quadratic equation in the variable 2. Two types of the quadratic equation are : Pure quadratic equation : The general form of the pure quadratic equation is Complete quadratic equation : The general form of the complete quadratic equation is 3. If The sum of the roots The product of the roots 4. The method of completing the square : Let , the quadratic equation
5. Quadratic Formula : The roots of a quadratic equation 6. The discrimanant of the quadratic equation 7. A quadratic equation (iv) If |
is an equation of the form 
, where 
are real numbers and 
. For example : 
etc .
, where 
are real numbers and 
,………. , etc .
are real numbers and 
,………. , etc .
and 
are the roots of the quadratic equation 
, then
.
or 
, provided 
.
.
,
,
.
, then the roots are reciprocal to each other .EXERCISE 4.1
1.Check whether the following are quadratic equations :
(i) 
(ii) 
(iii) 
(iv) 
(v) 
(vi) 
(vii) 
(viii) 
Solution : (i)
It is of the form 
So, the given equation is a quadratic equation .
(ii)
It is of the form
So, the given equation is a quadratic equation .
(iii) 
It is not of the form
So, the given equation is not a quadratic equation .
(iv)
It is of the form
So, the given equation is a quadratic equation .
(v)
It is of the form
So, the given equation is a quadratic equation .
(vi)
It is of the form
So, the given equation is a quadratic equation .
(vii)
It is not of the form
So, the given equation is not a quadratic equation .
(viii)
It is of the form
So, the given equation is a quadratic equation .
2. Represent the following situations in the form of quadratic equations :
(i) The area of a rectangular plot is 528 
. The length of the plot (in metres) is one more than twice its breadth . We needed to find the length and breadth of the plot .
Solution : Let 
be the breadth of the plot and the length of the plot will be 
.(in metres)
A/Q , 
= 0
Therefore , the given equation is a quadratic equation .
(ii) The product of two consecutive positive integers is 306 . We need to find the integers .
Solution: Let and 
are two consecutive positive integers respectively .
A/Q, 
Therefore, the given equation is a quadratic equation .
(iii) 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 .
Solution : Let be the present age of Rohan and Rohan’s mother age will be 
years .
Again , 3 years from now , the age of Rohan and Rohan’s mother will be 
and 
years respectively .
A/Q, 
Therefore, the given equation is a quadratic equation .
(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 .
Solution : Let, (in km/hrs) be the speed of the train .
A/Q, 
Therefore, the given equation is a quadratic equation .
EXERCISE 4.2
1. Find the roots of the following quadratic equations by factorization :
(i) 
Solution : We have,
So, 
or 
Therefore, the roots are 5 and 
.
(ii) 
Solution : We have ,
So, or 
Therefore, the roots are and 
.
(iii) 
Solution : we have , 
So, 
or 
Therefore , the roots are 
and 
.
(iv) 
Solution : We have , 
So, 
or 
Therefore, the roots are 
and .
(v) 
Solution : We have ,
or 
Therefore, the roots of 
and .
2. Solve the problems :
(i) John and Jivanti 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 .
Solution : Let be the marbles of John and Jivanti’s marbles will be 
. If 5 marbles has lost , then John and Jivanti’s marble will be 
and 
respectively .
A/Q, 
or 
Required the solutions are 9 and 36 .
(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 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 .
Solution : Let be the number of toys produced on that day and the cost of production of each that day Rs 
.
A/Q, 
or 
So, the number of toys is 25 or 30 .
3. Find two numbers whose sum is 27 and product is 182 .
Solution : Let be the one number and other number will be 
.
A/Q, 
or 
So, the two number are 13 and 14 .
4. Find two consecutive positive integers, sum of whose squares is 365 .
Solution: Let and 
are two consecutive positive integers respectively .
A/Q , 
or 
So, the two consecutive positive integers are 13 and 14 (= 13+1) .
5. The altitude of a right triangle is 7 cm less than its base . If the hypotenuse is 13 cm , find the other two sides .
Solution : Let (in cm) be the altitude of a right triangle and the base is 
cm .
A/Q, 
or 
(impossible)
So, the two sides of the triangle are 12 cm and 5 cm (12 – 7 = 5) .
6. A cottage industry produces a certain number of pottery articles in a day . If was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day . If the total cost of production on that day was Rs 90 , find the number of articles produced and the cost of each article .
Solution: Let be the number of articles produced in a day and the cost of product is Rs 
.
A/Q , 
or 
(impossible)
So, the number of articles produced in a day is 6 and the cost of each item is Rs 15 (2×6 + 3 = 15) .
EXERCISE 4.3
1. Find the roots of the following quadratic equations, if they exist , by the method of completing the square :
(i) 
Solution : (i) We have, 
Here , 
Now ,
So, the roots are 3 and 
.
(ii) 
Solution : We have,
Here , 
Now ,
Therefore, the roots of given quadratic equation are 
and 
.
(iii) 
Solution : We have ,
Here , 
Now ,
Therefore, the roots of given equation is 
and .
(iv) 
Solution : We have ,
Here , 
Since the square of a real number cannot be negative , therefore the quadratic equation has no real value .
2. Find the roots of the quadratic equation by applying the quadratic formula :
(i)
Solution : We have,
Here ,
We know that , 
Therefore, the roots of the given equation are 3 and .
(ii)
Solution : We have,
Here ,
We know that,
Therefore, the roots of the given equation are 
and 
.
(iii)
Solution : We have,
Here ,
We know that ,
Therefore , the roots of given equation are 
and .
(iv)
Solution : We have,
Here ,
We know that ,
Therefore, the roots of given equation are 
and .
3. Find the roots of the following equations :
(i) 
(ii) 
Solution : (i) We have,
Here , 
We know that,
Therefore, the roots of the given equation are 
and 
(ii) 
Solution : (ii) We have, 
or
Therefore , the roots of the equations are 1 and 2 .
4. The sum of the reciprocals of Rehman’s ages, (in years) 3 years ago and 5 years from now is 
. Find his present age .
Solution : Let (in years)be present age of Rehman .
3 years ago and 5 years from now , the age of Rehman will be 
and 
yrs respectively .
A/Q, 
Or 
Therefore, the roots of the equation is 7 and – 3 .
5. 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 .
Solution : Let be the marks in Mathematics of Shefali and her English marks will be 
.
A/Q , 
or 
Therefore, the marks obtained by Shefali is 12 or 18 and 13 or 17 respectively .
6. The diagonal of a rectangular field is 60 m more than the shorter side . If the longer side is 30 m more than the shorter side, find the sides of the field .
Solution : let, be the shorter side of a rectangular field and the longer side will be 
m
Therefore, the diagonal of a rectangular field is 
m .
A/Q , 
and 
(Impossible)
Thus, the shorter side of a rectangular field is 90 m and the longer side is 
.
7. The difference of square of two numbers is 180 . The square of the smaller number is 8 time the larger number . Find the two numbers .
Solution : Let and 
be two numbers .
A/Q , 
And 
From 
and 
, we get 
or 
[Impossible]
Putting 
in , we have 
Therefore, the numbers are 18 , 12 and 18 , – 12 .
8. A train travels 360 km at a uniform speed . If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey . Find the speed of the train .
Solution : Let (in km/h) be the speed of the train .
A/Q , 
[Impossible] or 
Therefore, the speed of the train is 40 km/h .
9. Two water taps together can fill a tank in 
hours . The tap of larger diameter takes 10 hours less than the smallest one to fill the tank separately . Find the time in which each tap can separately fill the tank .
Solution: let 
be the time taken by larger diameter tap and 
be the time taken by smaller diameter tap.
A/Q, 
Therefore, 
or 
[ 
]
Therefore, the time taken by larger diameter tap is 25 hours and the time taken by smaller diameter tap is 15 hours .
10. 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 11 km/h more than that of the passenger train, find the average speed of the two trains .
Solution : Let (in km/h) be the speed of the express train and the average speed of the express train is 
km/h .
A/Q , 
or 
[Impossible]
Therefore, the speed of the express train is 44 km/h and the average speed of the express train is 33 km/h . [ 
] .
11. Sum of the areas of two squares is 468 . If the difference of their perimeter is 24 m , find the sides of the two squares
Solution : Let and 
be the sides of the two squares respectively .
A/Q , 
And 
[ From 
]
[imposible] or 
Putting the value of 
in equation , we get 
Therefore, the sides of the two squares are 18 m and 12 m respectively .
EXERCISE 4.4
1. Find the nature of the roots of the following quadratic equations . If the real roots exists, find them :
(i) 
Solution : We have,
Here , 
Therefore, the equation has no real roots .
(ii) 
Solution : We have ,
Here , 
Therefore, the roots of the equation are equal .
Using quadratic formula , we have
or 
The roots of the equation are and .
(iii) 
Solution : We have,
Here , 
Therefore, the roots of the equation has two distinct real roots .
Using quadratic formula , we have
or 
The roots of the equation are 
and .
2. Find the values of 
for each of the following quadratic equations, so that they have two equal roots .
(i) 
Solution : We have ,
Here, 
Therefore, the value of is 
.
(ii) 
Solution : We have, 
Here , 
[ impossible] or 
Therefore, the value of is 6 .
3. Is it possible to design a rectangular mango grove whose length is twice its breadth , and the area is 800 ? If so, find its length and breadth .
Solution : Let and 
be the length and breadth of the rectangular mango grove respectively.
A/Q , 
And 
[ From (i)]
[ Only positive value]
Putting the value of 
in , we get 
Yes . So, the length and breadth of rectangular design is 40 m and 20 m respectively .
4. Is the following situation possible ? If so, determine their present ages .The sum of the ages of two friends is 20 years . Four years ago, the product of their ages in years was 48 .
Solution : Let (in years) be the present age of one friend and other friend age will be 
years .
Four years ago , the two friend age will be 
and 
years respectively .
A/Q, 
Here, 
So, the given equation has no real root . Therefore, the given situation is not possible .
5. Is it possible to design a rectangular park of perimeter 80 m and area 400 ? If so, find its length and breadth .
Solution : Let and be the length and breadth of the rectangular park .
A/Q, 
And 
[ From (i)]
or 
Putting the value of in 
, we get 
Yes . So, the length and breadth of the park is 20 m and 20 m respectively .
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