0
 is a quadratic equation in the variable x. Here a, b, c are real numbers and a ≠ 0
If α is the root of quadratic equation Quadratic Equation 2
Then Quadratic Equation 3

Quadratic Formula :

For a given quadratic equation, Quadratic Equation 1The roots can be given by
Quadratic Equation 1

Exercise 4.1 (NCERT Solution)

1. Check whether the following are quadratic equations:
Quadratic Equation 5
Since, the equation is in the form of Quadratic Equation 6 So, it is a quadratic equation.
Quadratic Equation 6
Since, the equation is in the form of Quadratic Equation 7 So, it is a quadratic equation.
Quadratic Equation 8
Since, the equation is not in the form of Quadratic Equation 6 So, it is not a quadratic equation.
Quadratic Equation 8
Since, the equation is in the form of Quadratic Equation 7 So, it is a quadratic equation.
Quadratic Equation 8
Since, the equation is in the form of Quadratic Equation 7 So, it is a quadratic equation.
Quadratic Equation 8
Since, the equation is not in the form of Quadratic Equation 7 So, it is not a quadratic equation.
Quadratic Equation 8
Since, the equation is not in the form of Quadratic Equation 7 So, it is not a quadratic equation.
Quadratic Equation 8
Since, the equation is in the form of Quadratic Equation 7 So, it is a quadratic equation.

Question: 2 – Represent the following situation in the form of quadratic equation:
(i) The area of a rectangular plot is 528 m2. 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.
Solution:
Quadratic Equation 8
Since, the equation is in the form of Quadratic Equation 7 So, it is a quadratic equation.
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
Solution:
Quadratic Equation 8
Since, the equation is in the form of Quadratic Equation 7 So, it 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 the Rohan’s age.
Solution:
Quadratic Equation 8
Since, the equation is in the form of Quadratic Equation 7 So, it is a quadratic equation.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8km/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:
Quadratic Equation 8 
Quadratic Equation 8
Since, the equation is in the form of Quadratic Equation 7 So, it is a quadratic equation.

Excercise - 4.2 (ncert) 

Question: 1 – Find the roots of the following quadratic equations by factorization:
QuadraticEquation1
QuadraticEquation2
QuadraticEquation3
QuadraticEquation4
QuadraticEquation5

Question: 2 – Solve the problems given in Example 1.
(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:
Given, John and Jivanti together have number of marbles = 45
After losing of 5 marbles each of them, number of marble = 45 – 5 – 5 = 45 – 10 = 35
QuadraticEquation6
QuadraticEquation7
(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:
QuadraticEquation8
Question: 3 – Find two numbers whose sum is 27 and product is 182.
Solution:
QuadraticEquation9

Question: 4 – Find two consecutive positive integers, sum of whose squares is 365.
Solution:
QuadraticEquation10
QuadraticEquation11
Question: 5 – The altitude of a right triangle is 7cm less than its base. If the hypotenuse is 13cm, find the other two sides.
Solution:
QuadraticEquation12
QuadraticEquation13
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 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:
QuadraticEquation14
QuadraticEquation15

Exercise - 4.3 

Question: 1 – Find the roots of the following quadratic equations, if they exists, by the method of completing square.
quadratic equation exercise 4.1_1
quadratic equation exercise 4.1_2
quadratic equation exercise 4.1_3
quadratic equation exercise 4.1_4
quadratic equation exercise 4.1_5
quadratic equation exercise 4.1_6
quadratic equation exercise 4.1_7
quadratic equation exercise 4.1_8
quadratic equation exercise 4.1_9

quadratic equation exercise 4.1_10
quadratic equation exercise 4.1_11
quadratic equation exercise 4.1_12
quadratic equation exercise 4.1_13
quadratic equation exercise 4.1_14

quadratic equation exercise 4.1_15
quadratic equation exercise 4.1_16
quadratic equation exercise 4.1_17
quadratic equation exercise 4.1_18
quadratic equation exercise 4.1_19
quadratic equation exercise 4.1_20
Now after discarding the negative value, we have x = 7
Thus, Rehman's present age = 7 years

Question: 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 us assume, marks in Mathematics = x
Therefore, marks in English = 30 – x
If she scores 2 marks more in Mathematics; then marks in mathematics = x +2
And if she scores 3 marks less in English, the marks in English = 30 – x – 3 = 27 – x
quadratic equation exercise 4.1_21
quadratic equation exercise 4.1_22
quadratic equation exercise 4.1_23
Question: 6 – The diagonal of a rectangular field is 60 meters more than the shorter side. If the longer side is 30 meters more than the shorter side, find the sides of the field.
Solution:
quadratic equation exercise 4.1_24
quadratic equation exercise 4.1_25
Question: 7 – The difference of squares of two numbers is 180. The square of the smaller number is 8 times the large number. Find the two numbers.
Solution:
quadratic equation exercise 4.1_26

Question: 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:
quadratic equation exercise 4.1_27
quadratic equation exercise 4.1_28
quadratic equation exercise 4.1_29
quadratic equation exercise 4.1_30
quadratic equation exercise 4.1_31
Question: 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:
quadratic equation exercise 4.1_31
quadratic equation exercise 4.1_31
quadratic equation exercise 4.1_31
Question: 11 – Sum of the areas of two squares is 468 square meter. If the difference of the perimeters is 24 m, find the sides of the two squares.
Solution:
quadratic equation exercise 4.1_31
quadratic equation exercise 4.1_31

Exercise - 4.4

Question: 1 – Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
quadratic equation exercise 4.4_1
quadratic equation exercise 4.4_2
quadratic equation exercise 4.4_3
Question: 2 – Find the value of k for each of the following quadratic equations, so that they have two equal roots.
quadratic equation exercise 4.4_4
quadratic equation exercise 4.4_5



Question: 3 – Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 square meter? If so, find its length and breadth.
Solution:
quadratic equation exercise 4.4_6
Question: 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:
quadratic equation exercise 4.4_7
Question: 5 – Is it possible to design a rectangular part of perimeter 80m and area 400 square meter? If so, find its length and breadth.
Solution:
quadratic equation exercise 4.4_8
quadratic equation exercise 4.4_9

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