Chapter 4. Quadratic Equations   

Chapter 4. Quadratic Equations

Exercise 4.1 complete solution Exercise 4.2 complete solution Exercise 4.3 complete solution

Important notes : 1. A quadratic equation in the variable  is an equation of the form , where  are real numbers and  . For example :  etc . 2. Two types of the quadratic equation are : Pure quadratic equation :  The general form of the pure quadratic equation is  , where are real numbers and  . For example :  ,………. , etc . Complete quadratic equation : The general form of the complete quadratic equation is  , where are real numbers and  . For example :  ,………. , etc . 3. If  and  are the roots of the quadratic equation  , then The sum of the roots The product of the roots 4. The method of completing the square : Let , the quadratic equation  .        or  5. Quadratic Formula :   The roots of a quadratic equation  are given by , provided  . 6. The discrimanant of the quadratic equation  is given by  . 7. A quadratic equation  has   (i) two distinct real roots, if   ,   (ii) two equal real(i.e., coincident) roots, if   ,   (iii) no real roots, if   .   (iv) If , then the roots are reciprocal to each other .

Class 10 Maths  Chapter 4. Quadratic Equations Exercise 4.1 Solutions

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 : (i) 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 : (ii)  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 : (iii)  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 .

Class 10 Maths  Chapter 4. Quadratic Equations Exercise 4.2 Solutions :

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)  .

Class 10 Maths  Chapter 4. Quadratic Equations Exercise 4.3 Solutions

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 k 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 

 [ 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 

 

  or

 

Putting the value of  in  , we get

     

Yes . So, the length and breadth of the park is 20 m and 20 m respectively .