05 - Question bank- Quadratic Equations - Class 10 - Maths Class 10 Notes | EduRev

Crash Course for Class 10 Maths by Let's tute

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Class 10 : 05 - Question bank- Quadratic Equations - Class 10 - Maths Class 10 Notes | EduRev

 Page 1


         QUADRATIC EQUATIONS 
 
 
 
FACTORISATION METHOD 
1) Solve: 0 4 3
2
? ? ? x x 
Solution: 
         0 4 ) 1 4 (
2
? ? ? ? x x 
               
1 , 4
0 ) 1 )( 4 (
0 ) 4 ( 1 ) 4 (
0 4 4
2
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
x x
x x
x x x
x x x
 
                 x = 4, x = -1 are solutions    
                     
     2)  Solve: 0 4
2
? ? x    
           Solution:  
                 Formula used is ) )( (
2 2
b a b a b a ? ? ? ? 
                   
2 , 2
0 ) 2 )( 2 (
0 2
2 2
? ? ? ?
? ? ? ?
? ?
x x
x x
x
 
                   x=2, x=-2 are solutions                         
 
 
    3)   Solve: 
0 2 3
2
? ? ? x x    
           Solution:  
                   0 2 ) 2 3 ( 3
2
? ? ? ? x x 
                   
1 ,
3
2
0 ) 1 )( 2 3 (
0 ) 1 ( 2 ) 1 ( 3
0 2 2 3 3
2
?
?
? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
x x
x x
x x x
x x x
 
                   x=
3
2 ?
, x=1 are solutions     
 
       
 
 
 
Page 2


         QUADRATIC EQUATIONS 
 
 
 
FACTORISATION METHOD 
1) Solve: 0 4 3
2
? ? ? x x 
Solution: 
         0 4 ) 1 4 (
2
? ? ? ? x x 
               
1 , 4
0 ) 1 )( 4 (
0 ) 4 ( 1 ) 4 (
0 4 4
2
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
x x
x x
x x x
x x x
 
                 x = 4, x = -1 are solutions    
                     
     2)  Solve: 0 4
2
? ? x    
           Solution:  
                 Formula used is ) )( (
2 2
b a b a b a ? ? ? ? 
                   
2 , 2
0 ) 2 )( 2 (
0 2
2 2
? ? ? ?
? ? ? ?
? ?
x x
x x
x
 
                   x=2, x=-2 are solutions                         
 
 
    3)   Solve: 
0 2 3
2
? ? ? x x    
           Solution:  
                   0 2 ) 2 3 ( 3
2
? ? ? ? x x 
                   
1 ,
3
2
0 ) 1 )( 2 3 (
0 ) 1 ( 2 ) 1 ( 3
0 2 2 3 3
2
?
?
? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
x x
x x
x x x
x x x
 
                   x=
3
2 ?
, x=1 are solutions     
 
       
 
 
 
                               
QUADRATIC EQUATIONS
 
 
 
 
 
 
 
     4)   Solve: 
3
2
2
1
1
?
?
?
?
?
?
a
a
a
a
 
          Solution: 
                  
3
2
2
1
1
?
?
?
?
?
?
a
a
a
a
 
                  
3
) 2 )( 1 (
) 1 )( 2 ( ) 2 )( 1 (
?
? ?
? ? ? ? ?
?
a a
a a a a
                 
               
2 , 5
0 ) 2 )( 5 (
0 ) 5 ( 2 ) 5 (
0 10 2 5
0 10 3
) 2 )( 1 ( 3 ) 1 )( 2 ( ) 2 )( 1 (
2
2
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ?
a a
a a
a a a
a a a
a a
a a a a a a
 
 
               a=-5, a=2 are solutions 
        
        
   COMPLETION OF SQUARE METHOD 
       
   5)  Solve: 5 ) 4 (
2
? ? x 
     Solution: 
  
? ?
5 4 5 4
5 4
5 ) 4 (
5 ) 4 (
2
2
? ? ? ?
? ?
? ? ?
? ?
orx x
x
x
x
 
            So 5 4 5 4 ? ? ? ? orx x are solutions. 
  
: 
6)   Solve: 0 7 6
2
? ? ? x x 
          Solution: 
                 7 6
2
? ? x x 
                 ..... 3 7 3 ) 3 )( ( 2
2 2 2
? ? ? ? x x (Adding (3
2
) to both side to make a perfect square) 
                 16 ) 3 (
2
? ? x 
        Taking Square root on both sides 
                 4 ) 3 ( ? ? ? x 
                 4 ) 3 ( 4 ) 3 ( ? ? ? ? ? x or x
 
                 
1 3 4 ? ? ? x
  Or 
7 3 4 ? ? ? ? ? x
 
 
                 7 , 1 ? ? x
 
(Taking square roots on both the side) 
Page 3


         QUADRATIC EQUATIONS 
 
 
 
FACTORISATION METHOD 
1) Solve: 0 4 3
2
? ? ? x x 
Solution: 
         0 4 ) 1 4 (
2
? ? ? ? x x 
               
1 , 4
0 ) 1 )( 4 (
0 ) 4 ( 1 ) 4 (
0 4 4
2
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
x x
x x
x x x
x x x
 
                 x = 4, x = -1 are solutions    
                     
     2)  Solve: 0 4
2
? ? x    
           Solution:  
                 Formula used is ) )( (
2 2
b a b a b a ? ? ? ? 
                   
2 , 2
0 ) 2 )( 2 (
0 2
2 2
? ? ? ?
? ? ? ?
? ?
x x
x x
x
 
                   x=2, x=-2 are solutions                         
 
 
    3)   Solve: 
0 2 3
2
? ? ? x x    
           Solution:  
                   0 2 ) 2 3 ( 3
2
? ? ? ? x x 
                   
1 ,
3
2
0 ) 1 )( 2 3 (
0 ) 1 ( 2 ) 1 ( 3
0 2 2 3 3
2
?
?
? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
x x
x x
x x x
x x x
 
                   x=
3
2 ?
, x=1 are solutions     
 
       
 
 
 
                               
QUADRATIC EQUATIONS
 
 
 
 
 
 
 
     4)   Solve: 
3
2
2
1
1
?
?
?
?
?
?
a
a
a
a
 
          Solution: 
                  
3
2
2
1
1
?
?
?
?
?
?
a
a
a
a
 
                  
3
) 2 )( 1 (
) 1 )( 2 ( ) 2 )( 1 (
?
? ?
? ? ? ? ?
?
a a
a a a a
                 
               
2 , 5
0 ) 2 )( 5 (
0 ) 5 ( 2 ) 5 (
0 10 2 5
0 10 3
) 2 )( 1 ( 3 ) 1 )( 2 ( ) 2 )( 1 (
2
2
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ?
a a
a a
a a a
a a a
a a
a a a a a a
 
 
               a=-5, a=2 are solutions 
        
        
   COMPLETION OF SQUARE METHOD 
       
   5)  Solve: 5 ) 4 (
2
? ? x 
     Solution: 
  
? ?
5 4 5 4
5 4
5 ) 4 (
5 ) 4 (
2
2
? ? ? ?
? ?
? ? ?
? ?
orx x
x
x
x
 
            So 5 4 5 4 ? ? ? ? orx x are solutions. 
  
: 
6)   Solve: 0 7 6
2
? ? ? x x 
          Solution: 
                 7 6
2
? ? x x 
                 ..... 3 7 3 ) 3 )( ( 2
2 2 2
? ? ? ? x x (Adding (3
2
) to both side to make a perfect square) 
                 16 ) 3 (
2
? ? x 
        Taking Square root on both sides 
                 4 ) 3 ( ? ? ? x 
                 4 ) 3 ( 4 ) 3 ( ? ? ? ? ? x or x
 
                 
1 3 4 ? ? ? x
  Or 
7 3 4 ? ? ? ? ? x
 
 
                 7 , 1 ? ? x
 
(Taking square roots on both the side) 
              QUADRATIC EQUATIONS
  
 
 
 
 
 
 
 
 
        So, x=1 and x=-7 are the solutions 
    
                 
 
         7) Solve:  
                   
0 5 2 4
2
? ? ? x x
 
              Solution: 
 
                   
5 2 4
2
? ? x x
 
              Dividing by 4 on both the sides 
 
           
4
5
2
1
2
? ? x x
 
           
                  
 
             
16
1
4
1
2
1
2
1
2
1
2 2 2
? ?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
? ? ?
?
?
?
?
?
?middleterm
                 
             Adding 
16
1
 both the sides to make a perfect square. 
                   
16
1
4
5
16
1
2
1
2
? ? ? ? x x
 
                  
16
21
4
1
2
? ?
?
?
?
?
?
? x
 
 
 
                  
4
21
16
21
4
1
? ? ? ? ?
?
?
?
?
?
? x
 
                  
4
21
4
1
? ? x
 
                  
4
21
4
1
4
21
4
1
? ? ? and x
 
 
           Therefore, 
4
21
4
1
4
21
4
1
? ? ? and x
 are the solutions. 
                 
                              
           
 
First we will make the coefficient of highest degree as 1  
We use this trick    
Taking square root on both the side 
Page 4


         QUADRATIC EQUATIONS 
 
 
 
FACTORISATION METHOD 
1) Solve: 0 4 3
2
? ? ? x x 
Solution: 
         0 4 ) 1 4 (
2
? ? ? ? x x 
               
1 , 4
0 ) 1 )( 4 (
0 ) 4 ( 1 ) 4 (
0 4 4
2
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
x x
x x
x x x
x x x
 
                 x = 4, x = -1 are solutions    
                     
     2)  Solve: 0 4
2
? ? x    
           Solution:  
                 Formula used is ) )( (
2 2
b a b a b a ? ? ? ? 
                   
2 , 2
0 ) 2 )( 2 (
0 2
2 2
? ? ? ?
? ? ? ?
? ?
x x
x x
x
 
                   x=2, x=-2 are solutions                         
 
 
    3)   Solve: 
0 2 3
2
? ? ? x x    
           Solution:  
                   0 2 ) 2 3 ( 3
2
? ? ? ? x x 
                   
1 ,
3
2
0 ) 1 )( 2 3 (
0 ) 1 ( 2 ) 1 ( 3
0 2 2 3 3
2
?
?
? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
x x
x x
x x x
x x x
 
                   x=
3
2 ?
, x=1 are solutions     
 
       
 
 
 
                               
QUADRATIC EQUATIONS
 
 
 
 
 
 
 
     4)   Solve: 
3
2
2
1
1
?
?
?
?
?
?
a
a
a
a
 
          Solution: 
                  
3
2
2
1
1
?
?
?
?
?
?
a
a
a
a
 
                  
3
) 2 )( 1 (
) 1 )( 2 ( ) 2 )( 1 (
?
? ?
? ? ? ? ?
?
a a
a a a a
                 
               
2 , 5
0 ) 2 )( 5 (
0 ) 5 ( 2 ) 5 (
0 10 2 5
0 10 3
) 2 )( 1 ( 3 ) 1 )( 2 ( ) 2 )( 1 (
2
2
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ?
a a
a a
a a a
a a a
a a
a a a a a a
 
 
               a=-5, a=2 are solutions 
        
        
   COMPLETION OF SQUARE METHOD 
       
   5)  Solve: 5 ) 4 (
2
? ? x 
     Solution: 
  
? ?
5 4 5 4
5 4
5 ) 4 (
5 ) 4 (
2
2
? ? ? ?
? ?
? ? ?
? ?
orx x
x
x
x
 
            So 5 4 5 4 ? ? ? ? orx x are solutions. 
  
: 
6)   Solve: 0 7 6
2
? ? ? x x 
          Solution: 
                 7 6
2
? ? x x 
                 ..... 3 7 3 ) 3 )( ( 2
2 2 2
? ? ? ? x x (Adding (3
2
) to both side to make a perfect square) 
                 16 ) 3 (
2
? ? x 
        Taking Square root on both sides 
                 4 ) 3 ( ? ? ? x 
                 4 ) 3 ( 4 ) 3 ( ? ? ? ? ? x or x
 
                 
1 3 4 ? ? ? x
  Or 
7 3 4 ? ? ? ? ? x
 
 
                 7 , 1 ? ? x
 
(Taking square roots on both the side) 
              QUADRATIC EQUATIONS
  
 
 
 
 
 
 
 
 
        So, x=1 and x=-7 are the solutions 
    
                 
 
         7) Solve:  
                   
0 5 2 4
2
? ? ? x x
 
              Solution: 
 
                   
5 2 4
2
? ? x x
 
              Dividing by 4 on both the sides 
 
           
4
5
2
1
2
? ? x x
 
           
                  
 
             
16
1
4
1
2
1
2
1
2
1
2 2 2
? ?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
? ? ?
?
?
?
?
?
?middleterm
                 
             Adding 
16
1
 both the sides to make a perfect square. 
                   
16
1
4
5
16
1
2
1
2
? ? ? ? x x
 
                  
16
21
4
1
2
? ?
?
?
?
?
?
? x
 
 
 
                  
4
21
16
21
4
1
? ? ? ? ?
?
?
?
?
?
? x
 
                  
4
21
4
1
? ? x
 
                  
4
21
4
1
4
21
4
1
? ? ? and x
 
 
           Therefore, 
4
21
4
1
4
21
4
1
? ? ? and x
 are the solutions. 
                 
                              
           
 
First we will make the coefficient of highest degree as 1  
We use this trick    
Taking square root on both the side 
       QUADRATIC EQUATIONS
  
 
 
 
 
 
 
 
  
     8) Solve: 
           
0 1 2
2
? ? ? x x
 
          Solution:  
         
1 2
2
? ? x x
 
        We use this trick 
? ? 1 1 2
2
1
2
1
2
2 2
? ? ? ?
?
?
?
?
?
? ? ? ?
?
?
?
?
?
?middleterm
                  
         Adding (1) on both the sides 
                 
1 1 1 2
2
? ? ? ? x x
  
                               
2 1 2 1
2 1
2 ) 1 (
2 ) 1 (
2
? ? ? ?
? ?
? ? ?
? ?
andx x
x
x
x
    
 
              
            So 2 1 2 1 ? ? ? or x are solutions. 
 
 
 
     
       DISCRIMINANT METHOD
  
          
    9) Solve:  
            
0 8 4
2
? ? ? x x
 
        Solution: 
          Here we use 
0
2
? ? ? c bx ax
general form of equation 
         We can write the given equation as 
          
0 ) 8 ( ) 4 ( ) 1 (
2
? ? ? ? ? ? x x
                           
          Here a = 1, b = -4 c = -8  
               
ac b D 4
2
? ?
 
                  
48
32 16
) 8 )( 1 ( 4 ) 4 (
2
?
? ?
? ? ? ?
 
           Therefore, D > 0 , since solution is, 
 
 
 
 
 
 
Page 5


         QUADRATIC EQUATIONS 
 
 
 
FACTORISATION METHOD 
1) Solve: 0 4 3
2
? ? ? x x 
Solution: 
         0 4 ) 1 4 (
2
? ? ? ? x x 
               
1 , 4
0 ) 1 )( 4 (
0 ) 4 ( 1 ) 4 (
0 4 4
2
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
x x
x x
x x x
x x x
 
                 x = 4, x = -1 are solutions    
                     
     2)  Solve: 0 4
2
? ? x    
           Solution:  
                 Formula used is ) )( (
2 2
b a b a b a ? ? ? ? 
                   
2 , 2
0 ) 2 )( 2 (
0 2
2 2
? ? ? ?
? ? ? ?
? ?
x x
x x
x
 
                   x=2, x=-2 are solutions                         
 
 
    3)   Solve: 
0 2 3
2
? ? ? x x    
           Solution:  
                   0 2 ) 2 3 ( 3
2
? ? ? ? x x 
                   
1 ,
3
2
0 ) 1 )( 2 3 (
0 ) 1 ( 2 ) 1 ( 3
0 2 2 3 3
2
?
?
? ?
? ? ? ?
? ? ? ? ?
? ? ? ?
x x
x x
x x x
x x x
 
                   x=
3
2 ?
, x=1 are solutions     
 
       
 
 
 
                               
QUADRATIC EQUATIONS
 
 
 
 
 
 
 
     4)   Solve: 
3
2
2
1
1
?
?
?
?
?
?
a
a
a
a
 
          Solution: 
                  
3
2
2
1
1
?
?
?
?
?
?
a
a
a
a
 
                  
3
) 2 )( 1 (
) 1 )( 2 ( ) 2 )( 1 (
?
? ?
? ? ? ? ?
?
a a
a a a a
                 
               
2 , 5
0 ) 2 )( 5 (
0 ) 5 ( 2 ) 5 (
0 10 2 5
0 10 3
) 2 )( 1 ( 3 ) 1 )( 2 ( ) 2 )( 1 (
2
2
? ? ? ?
? ? ? ?
? ? ? ? ?
? ? ? ? ?
? ? ? ?
? ? ? ? ? ? ? ? ?
a a
a a
a a a
a a a
a a
a a a a a a
 
 
               a=-5, a=2 are solutions 
        
        
   COMPLETION OF SQUARE METHOD 
       
   5)  Solve: 5 ) 4 (
2
? ? x 
     Solution: 
  
? ?
5 4 5 4
5 4
5 ) 4 (
5 ) 4 (
2
2
? ? ? ?
? ?
? ? ?
? ?
orx x
x
x
x
 
            So 5 4 5 4 ? ? ? ? orx x are solutions. 
  
: 
6)   Solve: 0 7 6
2
? ? ? x x 
          Solution: 
                 7 6
2
? ? x x 
                 ..... 3 7 3 ) 3 )( ( 2
2 2 2
? ? ? ? x x (Adding (3
2
) to both side to make a perfect square) 
                 16 ) 3 (
2
? ? x 
        Taking Square root on both sides 
                 4 ) 3 ( ? ? ? x 
                 4 ) 3 ( 4 ) 3 ( ? ? ? ? ? x or x
 
                 
1 3 4 ? ? ? x
  Or 
7 3 4 ? ? ? ? ? x
 
 
                 7 , 1 ? ? x
 
(Taking square roots on both the side) 
              QUADRATIC EQUATIONS
  
 
 
 
 
 
 
 
 
        So, x=1 and x=-7 are the solutions 
    
                 
 
         7) Solve:  
                   
0 5 2 4
2
? ? ? x x
 
              Solution: 
 
                   
5 2 4
2
? ? x x
 
              Dividing by 4 on both the sides 
 
           
4
5
2
1
2
? ? x x
 
           
                  
 
             
16
1
4
1
2
1
2
1
2
1
2 2 2
? ?
?
?
?
?
?
? ?
?
?
?
?
?
?
?
?
?
?
?
?
?
? ?
? ? ?
?
?
?
?
?
?middleterm
                 
             Adding 
16
1
 both the sides to make a perfect square. 
                   
16
1
4
5
16
1
2
1
2
? ? ? ? x x
 
                  
16
21
4
1
2
? ?
?
?
?
?
?
? x
 
 
 
                  
4
21
16
21
4
1
? ? ? ? ?
?
?
?
?
?
? x
 
                  
4
21
4
1
? ? x
 
                  
4
21
4
1
4
21
4
1
? ? ? and x
 
 
           Therefore, 
4
21
4
1
4
21
4
1
? ? ? and x
 are the solutions. 
                 
                              
           
 
First we will make the coefficient of highest degree as 1  
We use this trick    
Taking square root on both the side 
       QUADRATIC EQUATIONS
  
 
 
 
 
 
 
 
  
     8) Solve: 
           
0 1 2
2
? ? ? x x
 
          Solution:  
         
1 2
2
? ? x x
 
        We use this trick 
? ? 1 1 2
2
1
2
1
2
2 2
? ? ? ?
?
?
?
?
?
? ? ? ?
?
?
?
?
?
?middleterm
                  
         Adding (1) on both the sides 
                 
1 1 1 2
2
? ? ? ? x x
  
                               
2 1 2 1
2 1
2 ) 1 (
2 ) 1 (
2
? ? ? ?
? ?
? ? ?
? ?
andx x
x
x
x
    
 
              
            So 2 1 2 1 ? ? ? or x are solutions. 
 
 
 
     
       DISCRIMINANT METHOD
  
          
    9) Solve:  
            
0 8 4
2
? ? ? x x
 
        Solution: 
          Here we use 
0
2
? ? ? c bx ax
general form of equation 
         We can write the given equation as 
          
0 ) 8 ( ) 4 ( ) 1 (
2
? ? ? ? ? ? x x
                           
          Here a = 1, b = -4 c = -8  
               
ac b D 4
2
? ?
 
                  
48
32 16
) 8 )( 1 ( 4 ) 4 (
2
?
? ?
? ? ? ?
 
           Therefore, D > 0 , since solution is, 
 
 
 
 
 
 
             QUADRATIC EQUAIONS
  
 
 
 
 
 
 
 
             
2
48 4
,
2
48 4
2
48 ) 4 (
,
2
48 ) 4 (
2
,
2
? ?
? ?
? ? ? ? ? ?
? ?
? ? ? ?
?
x
x
a
D b
a
D b
x
 
                 
2
3 16 4
,
2
3 16 4 ? ? ? ?
? ? x
                
                
3 2 2 , 3 2 2
2
3 4 4
,
2
3 4 4
? ? ? ?
? ?
? ?
x
x
 
 
              So 3 2 2 3 2 2 ? ? ? or x are solutions. 
    
 
   
 
  10) Solve:  
            
4 3
2
? ? x x
 
 
         
Solution: 
           Here we use 
0
2
? ? ? c bx ax
general form of equation 
          We can write the given equation as 
          
0 ) 4 ( ) 3 ( ) 1 (
2
? ? ? ? ? x x
                       
          By comparing thye equation with general form we get,          
                
          Here a = 1, b = 3 c = -4  
              
ac b D 4
2
? ?
 
                 
25
16 9
) 4 )( 1 ( 4 ) 3 (
2
?
? ?
? ? ?
 
          Therefore, D > 0 , since solution is, 
 
 
 
 
 
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