# Class 10 Quadratic Equations - Solutions of Cbse Ncert Mathematics Textbook Exercise 4.2

## CBSE Board - Class X, Mathematics (SA-II / Term II)

#### NCERT Solutions for Mathematics Textbook Exercise 4.2

(NCERT Math Textbook Page 76)
1: Find the roots of the following quadratic equations by factorization:

Solution.1(i):
We have, x2 – 3x – 10 = 0
=> x2 + 2x – 5x – 10 = 0
=> x(x + 2) – 5(x + 2) = 0
=> (x + 2) (x – 5) = 0
Hence, the roots of this equation are,
(x + 2) = 0
or, x = –2 and
(x – 5) = 0
or,  x = 5
or, x = –2, 5
Solution.1(ii):
We have, 2x2 + x – 6 = 0
=> 2x2 + 4x – 3x – 6 = 0
=> 2x(x + 2) – 3(x + 2) = 0
=> (x + 2) (2x – 3) = 0
Hence, the roots of this equation are,

Solution.1(iii): Solution.1(iv): Solution.1(v): Taking hint from the above solve it yourself.

2: Represent the following situations mathematically:
(i) Jhon and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of the marbles they now have is 124. We would like to find out how many marbles they had to start with.
(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.2(i):
Let the number of marbles John had = x.
Then the number of marbles Jivanti had = (45 – x)
After losing 5 marbles by each of them, John has (x – 5) marbles and Jivanti has (40 – x) marbles.
As per the given condition,
(x – 5) (40 – x) = 124
=> x(40 – x) – 5(40 – x) = 124
=> 40x – x2 –200 + 5x = 124
=> –x2 + 45x – 324 = 0
=> –x2 + 36x + 9x – 324 = 0
=> (x – 36) (–x + 9) = 0
or, x = 36 and 9.
Therefore, the number of marbles John had is 36 and the number of marbles Jivanti had is
(45 – 9) i.e. 9. And if  the number with John is 9, then it will be 36 with Jivanti.

Solution.2(ii): Taking hint from 2(i) solution, try to solve it yourself.

3: Find two numbers whose sum is 27 and product is 182.
Solution.3: Let the first number be ‘x’ and the other number will be (27 – x).
As per the given condition,
x(27 – x) = 182
=> 27x – x2 = 182
=> –x2 + 27x – 182 = 0
=> –x2 + 14x + 13x – 182 = 0
=> –x(x – 14) + 13(x – 14) = 0
=> (13 – x) (x – 14) = 0
Hence, one number is 13 and the other number is (27 – 13) = 14.

4: Find two consecutive positive integers, sum of whose squares is 365.
Solution.4: Let the first positive integer be ‘x’ and the second positive integer will be then (x + 1). As per the given condition,
x2 + (x + 1)2 = 365
=> 2x2 + 2x – 364 = 0
=> x2 + x – 182 = 0
=> x2 – 13x + 14x – 182 = 0
=> x(x – 13) + 14(x – 13) = 0
=> (x + 14) (x – 13) = 0
or, x = 13 and –14.
Since, ‘x’ is the positive integer so it can not be –14. Hence, first positive integer is 13 and the second positive integer is (13 + 1) i.e. 14.

5: The altitude of a right triangle is 7cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
Solution.5:
Let the base be ‘x cm’ and the altitude is then (x – 7) cm. As per the given condition,
(13)2 = (x – 7)2 + x2
=> 169 = (x2 – 14x + 49) + x2
=> 2x2 – 14x – 120 = 0
=> x2 – 7x – 60 = 0
=> (x – 12) (x + 5) = 0
or, x = 12 and –5.
Since ‘x’ is base (one side) of a triangle, so it can not be negative. Hence, base is 12 cm and the other side (altitude) is (12 – 7) i.e. 5 cm.

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.
Hint to solve - Let the number of articles produced be ‘x’ and the cost of production of each article on that day (3 + 2x). Now as per the given condition, x(3 + 2x) = 90. Also, remember since ‘x’ is the number of the articles produced and so it can not be a negative number.

At the end of completing all Exercises from each Chapter given in NCERT Math textbook, we shall provide CBSE Notes (Hints) / important Solutions of CBSE Questions, Mathematics Sample Questions with their Solutions and many more…
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