Textbook: Quadratic Equations | Mathematics Class 10 (Maharashtra SSC Board) PDF Download

 Page 1


30
2 Quadratic Equations
· Quadratic equation : Introduction          ·  Methods of solving quadratic equation
· Nature of roots of quadratic equation  ·  Relation between roots and coefficients  
·  Applications of quadratic equations
Let’s recall.
You have studied polynomials last year. You know types of polynomials according to 
their degree. When the degree of polynomial is 1 it is called a linear polynomial and 
if degree of a polynomial is 2 it is called a quadratic polynomial. 
 Activity : Classify the following polynomials as linear and quadratic. 
   5x + 9,    x
2
 + 3x -5,   3x - 7,  3x
2 
- 5x,  5x
2
    Linear polynomials          Quadratic polynomials
 
 Now equate the quadratic polynomial to 0 and study the equation we get. Such 
type of equation is known as quadratic equation. In practical life we may use quadratic 
equations many times. 
 Ex. Sanket purchased a rectangular plot having area 200 m
2
. Length of the plot was 
10 m more than its breadth. Find the length and the breadth of the plot. 
 Let the breadth of the plot be  x metre.
 \ Length = (x + 10) metre
 Area of rectangle = length ´ breadth
 \ 200 = (x + 10) ´ x
 \ 200 = x
2
 + 10 x
 That is  x
2
 + 10x = 200
 \   x
2
 + 10x - 200 = 0
Let’s study.
Page 2


30
2 Quadratic Equations
· Quadratic equation : Introduction          ·  Methods of solving quadratic equation
· Nature of roots of quadratic equation  ·  Relation between roots and coefficients  
·  Applications of quadratic equations
Let’s recall.
You have studied polynomials last year. You know types of polynomials according to 
their degree. When the degree of polynomial is 1 it is called a linear polynomial and 
if degree of a polynomial is 2 it is called a quadratic polynomial. 
 Activity : Classify the following polynomials as linear and quadratic. 
   5x + 9,    x
2
 + 3x -5,   3x - 7,  3x
2 
- 5x,  5x
2
    Linear polynomials          Quadratic polynomials
 
 Now equate the quadratic polynomial to 0 and study the equation we get. Such 
type of equation is known as quadratic equation. In practical life we may use quadratic 
equations many times. 
 Ex. Sanket purchased a rectangular plot having area 200 m
2
. Length of the plot was 
10 m more than its breadth. Find the length and the breadth of the plot. 
 Let the breadth of the plot be  x metre.
 \ Length = (x + 10) metre
 Area of rectangle = length ´ breadth
 \ 200 = (x + 10) ´ x
 \ 200 = x
2
 + 10 x
 That is  x
2
 + 10x = 200
 \   x
2
 + 10x - 200 = 0
Let’s study.
31
 Now, solving equation x
2
 + 10x - 200 = 0, we will decide the dimensions of the 
plot. 
 Let us study how to solve the quadratic equation. 
    
 
 
Let’s learn.
 Standard form of quadratic equation
 The equation involving one variable with all indices as whole numbers and having 
2 as the maximum index of the variable is called the quadratic equation. 
 General form is  ax
2
 + bx + c = 0 
 In ax
2
 + bx + c = 0,   a, b, c are real numbers and a ¹ 0.
  ax
2
 + bx + c = 0 is the general form of quadratic equation. 
Activity : Complete the following table
Quadratic 
Equation
General form a b c
x
2
 - 4 = 0
x
2
 + 0x - 4 = 0
1 0 -4
 y
2
 = 2y - 7 . . .  . . .  . . . . . . . . . . . .
x
2
 + 2x = 0 . . .  . . .  . . . . . . . . . . . .
ÒÒÒ??? Solved Examples ÒÒÒ???
Ex. (1)  Decide which of the following are quadratic equations ?
 (1) 3x
2
 - 5x + 3 = 0  (2) 9y
2
 + 5= 0  (3) m
3
 - 5m
2
 + 4 = 0 (4) (l + 2) (l - 5) = 0
Solution : (1) In the equation 3x
2
 - 5x + 3 = 0, x is the only variable and maximum  
   index of the variable  is 2
  \ It is a quadratic equation. 
   
Activity :
 
 x
2
 + 3x -5,  3x
2 
- 5x, 5x
2
; Write the polynomials in the index form. 
Observe the coefficients and fill in the boxes. 
  x
2
 + 3x -5 ,   3x
2 
- 5x + 0 ,  5x
2 
+ 0x + 0
 
 
?? Coefficients of  x
2  
are   1 ,  3  and  5 these coefficients are non zero.
?? Coefficients of   x are     3,   
 
   and   respectively. 
?? Constants terms are  ,      and    respectively. 
 Here constant term of second and third polynomial is zero.
Let’s recall.
Page 3


30
2 Quadratic Equations
· Quadratic equation : Introduction          ·  Methods of solving quadratic equation
· Nature of roots of quadratic equation  ·  Relation between roots and coefficients  
·  Applications of quadratic equations
Let’s recall.
You have studied polynomials last year. You know types of polynomials according to 
their degree. When the degree of polynomial is 1 it is called a linear polynomial and 
if degree of a polynomial is 2 it is called a quadratic polynomial. 
 Activity : Classify the following polynomials as linear and quadratic. 
   5x + 9,    x
2
 + 3x -5,   3x - 7,  3x
2 
- 5x,  5x
2
    Linear polynomials          Quadratic polynomials
 
 Now equate the quadratic polynomial to 0 and study the equation we get. Such 
type of equation is known as quadratic equation. In practical life we may use quadratic 
equations many times. 
 Ex. Sanket purchased a rectangular plot having area 200 m
2
. Length of the plot was 
10 m more than its breadth. Find the length and the breadth of the plot. 
 Let the breadth of the plot be  x metre.
 \ Length = (x + 10) metre
 Area of rectangle = length ´ breadth
 \ 200 = (x + 10) ´ x
 \ 200 = x
2
 + 10 x
 That is  x
2
 + 10x = 200
 \   x
2
 + 10x - 200 = 0
Let’s study.
31
 Now, solving equation x
2
 + 10x - 200 = 0, we will decide the dimensions of the 
plot. 
 Let us study how to solve the quadratic equation. 
    
 
 
Let’s learn.
 Standard form of quadratic equation
 The equation involving one variable with all indices as whole numbers and having 
2 as the maximum index of the variable is called the quadratic equation. 
 General form is  ax
2
 + bx + c = 0 
 In ax
2
 + bx + c = 0,   a, b, c are real numbers and a ¹ 0.
  ax
2
 + bx + c = 0 is the general form of quadratic equation. 
Activity : Complete the following table
Quadratic 
Equation
General form a b c
x
2
 - 4 = 0
x
2
 + 0x - 4 = 0
1 0 -4
 y
2
 = 2y - 7 . . .  . . .  . . . . . . . . . . . .
x
2
 + 2x = 0 . . .  . . .  . . . . . . . . . . . .
ÒÒÒ??? Solved Examples ÒÒÒ???
Ex. (1)  Decide which of the following are quadratic equations ?
 (1) 3x
2
 - 5x + 3 = 0  (2) 9y
2
 + 5= 0  (3) m
3
 - 5m
2
 + 4 = 0 (4) (l + 2) (l - 5) = 0
Solution : (1) In the equation 3x
2
 - 5x + 3 = 0, x is the only variable and maximum  
   index of the variable  is 2
  \ It is a quadratic equation. 
   
Activity :
 
 x
2
 + 3x -5,  3x
2 
- 5x, 5x
2
; Write the polynomials in the index form. 
Observe the coefficients and fill in the boxes. 
  x
2
 + 3x -5 ,   3x
2 
- 5x + 0 ,  5x
2 
+ 0x + 0
 
 
?? Coefficients of  x
2  
are   1 ,  3  and  5 these coefficients are non zero.
?? Coefficients of   x are     3,   
 
   and   respectively. 
?? Constants terms are  ,      and    respectively. 
 Here constant term of second and third polynomial is zero.
Let’s recall.
32
 (2) In the equation 9y
2
 + 5= 0,  is the only variable and maximum index of the 
  variable is 
  \It  a quadratic equation.
 (3) In the equation m
3
 - 5m
2
 + 4 = 0, is the only variable but maximum index 
  of the variable is not 2.
  \It  a quadratic equation.
 (4) (l + 2) (l - 5) = 0 
  \ l (l - 5) + 2 (l - 5) = 0 
  \ l
2
 - 5l + 2l - 10 = 0  
  \ l
2
 - 3l - 10 = 0, In this equation  is the only variable and maximum index 
  of the variable is  .
  \It  a quadratic equation.
Let’s learn.
 Roots of a quadratic equation
 In the previous class you have studied that if value of the polynomial is zero for 
x = a then (x - a) is a factor of that polynomial. That is if p(x) is a polynomial and  
p(a) = 0 then (x - a) is a factor of p(x). In this case ’a’ is the root or solution of  p(x) = 0 
For Example , 
 Let x = -6 in the polynomial x
2
 + 5x - 6
      x
2
 + 5x - 6 = (-6)
2
 + 5 ´ (-6) - 6 
     = 36 - 30 - 6    = 0
 \ x = -6 is a solution of the equation.  
 Hence -6 is one root of the equation 
 x
2
 + 5x - 6 = 0  
Let x = 2 in polynomial x
2
 + 5x - 6  
     x
2
 + 5x - 6 = 2
2
 + 5 ´ 2 - 6 
     = 4 + 10 - 6
     =  8 ¹ 0
\ x = 2 is not a solution of the   
  equation  x
2
 + 5x - 6 = 0 
ÒÒÒ?? Solved Example ÒÒÒ??
Ex.    2x
2
 - 7x + 6 = 0 check whether  (i) x = 
3
2
 , (ii) x = -2 are solutions of the equations. 
Solution : (i) Put x = 
3
2
 in the polynomial 2x
2
 - 7x + 6  
    2x
2
 - 7x + 6 = 2
3
2
2






 - 7
3
2





 + 6 
Page 4


30
2 Quadratic Equations
· Quadratic equation : Introduction          ·  Methods of solving quadratic equation
· Nature of roots of quadratic equation  ·  Relation between roots and coefficients  
·  Applications of quadratic equations
Let’s recall.
You have studied polynomials last year. You know types of polynomials according to 
their degree. When the degree of polynomial is 1 it is called a linear polynomial and 
if degree of a polynomial is 2 it is called a quadratic polynomial. 
 Activity : Classify the following polynomials as linear and quadratic. 
   5x + 9,    x
2
 + 3x -5,   3x - 7,  3x
2 
- 5x,  5x
2
    Linear polynomials          Quadratic polynomials
 
 Now equate the quadratic polynomial to 0 and study the equation we get. Such 
type of equation is known as quadratic equation. In practical life we may use quadratic 
equations many times. 
 Ex. Sanket purchased a rectangular plot having area 200 m
2
. Length of the plot was 
10 m more than its breadth. Find the length and the breadth of the plot. 
 Let the breadth of the plot be  x metre.
 \ Length = (x + 10) metre
 Area of rectangle = length ´ breadth
 \ 200 = (x + 10) ´ x
 \ 200 = x
2
 + 10 x
 That is  x
2
 + 10x = 200
 \   x
2
 + 10x - 200 = 0
Let’s study.
31
 Now, solving equation x
2
 + 10x - 200 = 0, we will decide the dimensions of the 
plot. 
 Let us study how to solve the quadratic equation. 
    
 
 
Let’s learn.
 Standard form of quadratic equation
 The equation involving one variable with all indices as whole numbers and having 
2 as the maximum index of the variable is called the quadratic equation. 
 General form is  ax
2
 + bx + c = 0 
 In ax
2
 + bx + c = 0,   a, b, c are real numbers and a ¹ 0.
  ax
2
 + bx + c = 0 is the general form of quadratic equation. 
Activity : Complete the following table
Quadratic 
Equation
General form a b c
x
2
 - 4 = 0
x
2
 + 0x - 4 = 0
1 0 -4
 y
2
 = 2y - 7 . . .  . . .  . . . . . . . . . . . .
x
2
 + 2x = 0 . . .  . . .  . . . . . . . . . . . .
ÒÒÒ??? Solved Examples ÒÒÒ???
Ex. (1)  Decide which of the following are quadratic equations ?
 (1) 3x
2
 - 5x + 3 = 0  (2) 9y
2
 + 5= 0  (3) m
3
 - 5m
2
 + 4 = 0 (4) (l + 2) (l - 5) = 0
Solution : (1) In the equation 3x
2
 - 5x + 3 = 0, x is the only variable and maximum  
   index of the variable  is 2
  \ It is a quadratic equation. 
   
Activity :
 
 x
2
 + 3x -5,  3x
2 
- 5x, 5x
2
; Write the polynomials in the index form. 
Observe the coefficients and fill in the boxes. 
  x
2
 + 3x -5 ,   3x
2 
- 5x + 0 ,  5x
2 
+ 0x + 0
 
 
?? Coefficients of  x
2  
are   1 ,  3  and  5 these coefficients are non zero.
?? Coefficients of   x are     3,   
 
   and   respectively. 
?? Constants terms are  ,      and    respectively. 
 Here constant term of second and third polynomial is zero.
Let’s recall.
32
 (2) In the equation 9y
2
 + 5= 0,  is the only variable and maximum index of the 
  variable is 
  \It  a quadratic equation.
 (3) In the equation m
3
 - 5m
2
 + 4 = 0, is the only variable but maximum index 
  of the variable is not 2.
  \It  a quadratic equation.
 (4) (l + 2) (l - 5) = 0 
  \ l (l - 5) + 2 (l - 5) = 0 
  \ l
2
 - 5l + 2l - 10 = 0  
  \ l
2
 - 3l - 10 = 0, In this equation  is the only variable and maximum index 
  of the variable is  .
  \It  a quadratic equation.
Let’s learn.
 Roots of a quadratic equation
 In the previous class you have studied that if value of the polynomial is zero for 
x = a then (x - a) is a factor of that polynomial. That is if p(x) is a polynomial and  
p(a) = 0 then (x - a) is a factor of p(x). In this case ’a’ is the root or solution of  p(x) = 0 
For Example , 
 Let x = -6 in the polynomial x
2
 + 5x - 6
      x
2
 + 5x - 6 = (-6)
2
 + 5 ´ (-6) - 6 
     = 36 - 30 - 6    = 0
 \ x = -6 is a solution of the equation.  
 Hence -6 is one root of the equation 
 x
2
 + 5x - 6 = 0  
Let x = 2 in polynomial x
2
 + 5x - 6  
     x
2
 + 5x - 6 = 2
2
 + 5 ´ 2 - 6 
     = 4 + 10 - 6
     =  8 ¹ 0
\ x = 2 is not a solution of the   
  equation  x
2
 + 5x - 6 = 0 
ÒÒÒ?? Solved Example ÒÒÒ??
Ex.    2x
2
 - 7x + 6 = 0 check whether  (i) x = 
3
2
 , (ii) x = -2 are solutions of the equations. 
Solution : (i) Put x = 
3
2
 in the polynomial 2x
2
 - 7x + 6  
    2x
2
 - 7x + 6 = 2
3
2
2






 - 7
3
2





 + 6 
33
(1) ax
2
 + bx + c = 0 is the general form of equation where  a, b, c are real 
numbers and ’a ’
 
is non zero.
(2) The values of variable which satisfy the equation [or the value for 
which both the sides of equation are equal] are called solutions or 
roots of the equation. 
     = 2  ´ 
9
4
 - 
21
2
+ 6           
    = 
9
2
 - 
21
2
+ 
12
2
  = 0
 \ x = 
3
2
 is a solution of the equation. 
 (ii) Let x = -2  in 2x
2
 - 7x + 6   
   2x
2
 - 7x + 6 = 2(-2)
2
 - 7(-2) + 6 
           = 2 ´ 4 + 14 + 6
           = 28 ¹ 0
 \ x = -2  is not a solution of the equation.
Activity : If x = 5 is  a root of equation kx
2
 - 14x - 5 = 0 then find the value of 
k  by completing the following activity. 
Solution : One of the roots of equation kx
2
 - 14x - 5 = 0 is   . 
  \ Now Let x =  in the equation.  
  k  
2
 -  14  - 5 = 0
  \  25k - 70 - 5 = 0
  25k -  = 0
  25k = 
 \  k =    = 3
Let’s remember!
Page 5


30
2 Quadratic Equations
· Quadratic equation : Introduction          ·  Methods of solving quadratic equation
· Nature of roots of quadratic equation  ·  Relation between roots and coefficients  
·  Applications of quadratic equations
Let’s recall.
You have studied polynomials last year. You know types of polynomials according to 
their degree. When the degree of polynomial is 1 it is called a linear polynomial and 
if degree of a polynomial is 2 it is called a quadratic polynomial. 
 Activity : Classify the following polynomials as linear and quadratic. 
   5x + 9,    x
2
 + 3x -5,   3x - 7,  3x
2 
- 5x,  5x
2
    Linear polynomials          Quadratic polynomials
 
 Now equate the quadratic polynomial to 0 and study the equation we get. Such 
type of equation is known as quadratic equation. In practical life we may use quadratic 
equations many times. 
 Ex. Sanket purchased a rectangular plot having area 200 m
2
. Length of the plot was 
10 m more than its breadth. Find the length and the breadth of the plot. 
 Let the breadth of the plot be  x metre.
 \ Length = (x + 10) metre
 Area of rectangle = length ´ breadth
 \ 200 = (x + 10) ´ x
 \ 200 = x
2
 + 10 x
 That is  x
2
 + 10x = 200
 \   x
2
 + 10x - 200 = 0
Let’s study.
31
 Now, solving equation x
2
 + 10x - 200 = 0, we will decide the dimensions of the 
plot. 
 Let us study how to solve the quadratic equation. 
    
 
 
Let’s learn.
 Standard form of quadratic equation
 The equation involving one variable with all indices as whole numbers and having 
2 as the maximum index of the variable is called the quadratic equation. 
 General form is  ax
2
 + bx + c = 0 
 In ax
2
 + bx + c = 0,   a, b, c are real numbers and a ¹ 0.
  ax
2
 + bx + c = 0 is the general form of quadratic equation. 
Activity : Complete the following table
Quadratic 
Equation
General form a b c
x
2
 - 4 = 0
x
2
 + 0x - 4 = 0
1 0 -4
 y
2
 = 2y - 7 . . .  . . .  . . . . . . . . . . . .
x
2
 + 2x = 0 . . .  . . .  . . . . . . . . . . . .
ÒÒÒ??? Solved Examples ÒÒÒ???
Ex. (1)  Decide which of the following are quadratic equations ?
 (1) 3x
2
 - 5x + 3 = 0  (2) 9y
2
 + 5= 0  (3) m
3
 - 5m
2
 + 4 = 0 (4) (l + 2) (l - 5) = 0
Solution : (1) In the equation 3x
2
 - 5x + 3 = 0, x is the only variable and maximum  
   index of the variable  is 2
  \ It is a quadratic equation. 
   
Activity :
 
 x
2
 + 3x -5,  3x
2 
- 5x, 5x
2
; Write the polynomials in the index form. 
Observe the coefficients and fill in the boxes. 
  x
2
 + 3x -5 ,   3x
2 
- 5x + 0 ,  5x
2 
+ 0x + 0
 
 
?? Coefficients of  x
2  
are   1 ,  3  and  5 these coefficients are non zero.
?? Coefficients of   x are     3,   
 
   and   respectively. 
?? Constants terms are  ,      and    respectively. 
 Here constant term of second and third polynomial is zero.
Let’s recall.
32
 (2) In the equation 9y
2
 + 5= 0,  is the only variable and maximum index of the 
  variable is 
  \It  a quadratic equation.
 (3) In the equation m
3
 - 5m
2
 + 4 = 0, is the only variable but maximum index 
  of the variable is not 2.
  \It  a quadratic equation.
 (4) (l + 2) (l - 5) = 0 
  \ l (l - 5) + 2 (l - 5) = 0 
  \ l
2
 - 5l + 2l - 10 = 0  
  \ l
2
 - 3l - 10 = 0, In this equation  is the only variable and maximum index 
  of the variable is  .
  \It  a quadratic equation.
Let’s learn.
 Roots of a quadratic equation
 In the previous class you have studied that if value of the polynomial is zero for 
x = a then (x - a) is a factor of that polynomial. That is if p(x) is a polynomial and  
p(a) = 0 then (x - a) is a factor of p(x). In this case ’a’ is the root or solution of  p(x) = 0 
For Example , 
 Let x = -6 in the polynomial x
2
 + 5x - 6
      x
2
 + 5x - 6 = (-6)
2
 + 5 ´ (-6) - 6 
     = 36 - 30 - 6    = 0
 \ x = -6 is a solution of the equation.  
 Hence -6 is one root of the equation 
 x
2
 + 5x - 6 = 0  
Let x = 2 in polynomial x
2
 + 5x - 6  
     x
2
 + 5x - 6 = 2
2
 + 5 ´ 2 - 6 
     = 4 + 10 - 6
     =  8 ¹ 0
\ x = 2 is not a solution of the   
  equation  x
2
 + 5x - 6 = 0 
ÒÒÒ?? Solved Example ÒÒÒ??
Ex.    2x
2
 - 7x + 6 = 0 check whether  (i) x = 
3
2
 , (ii) x = -2 are solutions of the equations. 
Solution : (i) Put x = 
3
2
 in the polynomial 2x
2
 - 7x + 6  
    2x
2
 - 7x + 6 = 2
3
2
2






 - 7
3
2





 + 6 
33
(1) ax
2
 + bx + c = 0 is the general form of equation where  a, b, c are real 
numbers and ’a ’
 
is non zero.
(2) The values of variable which satisfy the equation [or the value for 
which both the sides of equation are equal] are called solutions or 
roots of the equation. 
     = 2  ´ 
9
4
 - 
21
2
+ 6           
    = 
9
2
 - 
21
2
+ 
12
2
  = 0
 \ x = 
3
2
 is a solution of the equation. 
 (ii) Let x = -2  in 2x
2
 - 7x + 6   
   2x
2
 - 7x + 6 = 2(-2)
2
 - 7(-2) + 6 
           = 2 ´ 4 + 14 + 6
           = 28 ¹ 0
 \ x = -2  is not a solution of the equation.
Activity : If x = 5 is  a root of equation kx
2
 - 14x - 5 = 0 then find the value of 
k  by completing the following activity. 
Solution : One of the roots of equation kx
2
 - 14x - 5 = 0 is   . 
  \ Now Let x =  in the equation.  
  k  
2
 -  14  - 5 = 0
  \  25k - 70 - 5 = 0
  25k -  = 0
  25k = 
 \  k =    = 3
Let’s remember!
34
Practice Set 2.1
1.  Write any two quadratic equations. 
2.  Decide which of the following are quadratic equations.
 (1) x
2
 + 5 x - 2 = 0  (2) y
2
 = 5 y - 10       (3) y
2
 + 
1
y
  = 2
 (4) x + 
1
x
 = -2            (5) (m + 2) (m - 5) = 0     (6) m
3
 + 3 m
2
 -2 = 3 m
3
3. Write the following equations in the form 
 
ax
2
 + bx + c = 0, then write the values of  
a, b, c for each equation. 
 (1) 2y =10 - y
2
   (2) (x - 1)
2
 = 2 x + 3      (3) x
2
 + 5x  = -(3 - x)
 (4) 3m
2
 =  2 m
2
 - 9  (5) P (3 + 6p) = -5          (6) x
2
 - 9 = 13
4.  Determine whether the values given against each of the quadratic equation are the 
roots of the equation.
 (1) x
2
 + 4x - 5 = 0 , x
 
= 1, -1   (2) 2m
2
 - 5m  = 0 ,  m = 2, 
5
2
5.  Find k if  x = 3 is a root of equation kx
2
 - 10x + 3 = 0 .
6. One of the roots of equation 5m
2
 + 2m + k = 0 is 
-7
5
. Complete the following activity 
to find the value of ’k’. 
Solution :  is a root of quadratic equation 5m
2
 + 2m + k = 0  
  \ Put m =   in the equation. 
   5 ´  
2
 + 2 ´  + k = 0 
    +  + k = 0 
    +  k = 0
         k =  
  
 Last year you have studied the methods to find the factors of quadratic polynomials 
like  x
2
 - 4x - 5, 2m
2
 - 5m, a
2
 - 25. Try the following activity and revise the same. 
 Activity : Find the factors of the following polynomials. 
 (1) x
2
 - 4 x - 5     (2) 2m
2
 - 5 m    (3) a
2
 - 25
 =  x
2
 - 5 x + 1x - 5             = . . .  . . .        = a
2
 - 5
2
 
= x (. . . .) +1(. . . .)            = (. . . .)  (. . . .)
 = (. . . .)  (. . . .)
Let’s recall.