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

Then

## Quadratic Formula :

For a given quadratic equation, The roots can be given by

### Exercise 4.1 (NCERT Solution)

1. Check whether the following are quadratic equations:

Since, the equation is in the form of So, it is a quadratic equation.

Since, the equation is in the form of So, it is a quadratic equation.

Since, the equation is not in the form of So, it is not a quadratic equation.

Since, the equation is in the form of So, it is a quadratic equation.

Since, the equation is in the form of So, it is a quadratic equation.

Since, the equation is not in the form of So, it is not a quadratic equation.

Since, the equation is not in the form of So, it is not a quadratic equation.

Since, the equation is in the form of 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:

Since, the equation is in the form of So, it is a quadratic equation.

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

Solution:

Since, the equation is in the form of 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:

Since, the equation is in the form of 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:

Since, the equation is in the form of So, it is a quadratic equation.

## Excercise - 4.2 (ncert)

Question: 1 – Find the roots of the following quadratic equations by factorization:

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

(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:

Question: 3 – Find two numbers whose sum is 27 and product is 182.

Solution:

Question: 4 – Find two consecutive positive integers, sum of whose squares is 365.

Solution:

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:

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:

## Exercise - 4.3

Question: 1 – Find the roots of the following quadratic equations, if they exists, by the method of completing square.

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

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:

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:

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:

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:

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:

## Exercise - 4.4

Question: 1 – Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:

Question: 2 – Find the value of k for each of the following quadratic equations, so that they have two equal roots.

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:

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:

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:

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