Important Quadratic Equation Formulas for JEE and NEET

 Page 1


Page # 22
QUADRATIC EQUATIONS
1. Quadratic Equation : a
 
x
2
 + b
 
x + c = 0,  a ? 0
x = 
a 2
c a 4 b b
2
? ? ?
, The expression b
2
 ? 4 a
 
c ? D is called
discriminant of quadratic equation.
If ?, ? are the roots, then (a) ? + ? = ? 
a
b
    (b) ? ? ? ? = 
a
c
A quadratic equation whose roots are ? & ?, is (x ? ? ?) (x ? ? ?)
= 0  i.e. x
2
 ? ( ? + ? ) x + ? ? ?  = 0
2. Nature of Roots:
Consider the quadratic equation, a
 
x
2
 + b
 
x + c = 0 having ?
,
 ? as its roots;
D ? b
2
 ? 4 a
 
c
      
    D = 0     D ? 0
Roots are equal ? = ? = ? b/2a Roots are unequal
  
   a, b, c ? R & D > 0 a, b, c ? R & D < 0
    Roots are real       Roots are imaginary ? = p + i
 
q,  ? = p ? i
 
q
     
a, b, c ? Q & a, b, c ? Q &
D is a perfect square   D is not a perfect square
?  Roots are rational ?  Roots are irrational
? i.e. ? = p +
q
, ? = p ?
 
q
a = 1, b, c ? ? & D is a perfect square
? Roots are integral.
Page 2


Page # 22
QUADRATIC EQUATIONS
1. Quadratic Equation : a
 
x
2
 + b
 
x + c = 0,  a ? 0
x = 
a 2
c a 4 b b
2
? ? ?
, The expression b
2
 ? 4 a
 
c ? D is called
discriminant of quadratic equation.
If ?, ? are the roots, then (a) ? + ? = ? 
a
b
    (b) ? ? ? ? = 
a
c
A quadratic equation whose roots are ? & ?, is (x ? ? ?) (x ? ? ?)
= 0  i.e. x
2
 ? ( ? + ? ) x + ? ? ?  = 0
2. Nature of Roots:
Consider the quadratic equation, a
 
x
2
 + b
 
x + c = 0 having ?
,
 ? as its roots;
D ? b
2
 ? 4 a
 
c
      
    D = 0     D ? 0
Roots are equal ? = ? = ? b/2a Roots are unequal
  
   a, b, c ? R & D > 0 a, b, c ? R & D < 0
    Roots are real       Roots are imaginary ? = p + i
 
q,  ? = p ? i
 
q
     
a, b, c ? Q & a, b, c ? Q &
D is a perfect square   D is not a perfect square
?  Roots are rational ?  Roots are irrational
? i.e. ? = p +
q
, ? = p ?
 
q
a = 1, b, c ? ? & D is a perfect square
? Roots are integral.
Page # 23
3. Common Roots:
Consider two quadratic equations a
1 
x
2
 + b
1 
x + c
1
 = 0 & a
2 
x
2
 + b
2 
x + c
2
 = 0.
(i) If two quadratic equations have both roots common, then
2
1
a
a
 =
2
1
b
b
 =
2
1
c
c
.
(ii) If only one root ? is common, then
? =
1 2 2 1
1 2 2 1
b a b a
a c a c
?
?
 = 
1 2 2 1
1 2 2 1
a c a c
c b c b
?
?
4. Range of Quadratic Expression f (x) = a
 
x
2
 + b
 
x + c.
Range in restricted domain: Given x ? [x
1
, x
2
]
(a) If ?
a 2
b
 ? [x
1
, x
2
] then,
f(x) ? ? ? ? ? ? ? ) ( ) ( ) ( ) (
2 1 2 1
x f , xf max , xf , xf min
(b) If ?
a 2
b
 ? [x
1
, x
2
] then,
f(x) ? 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
a 4
D
, xf , xf max ,
a 4
D
, xf , xf min ) ( ) ( ) ( ) (
21 21
Page 3


Page # 22
QUADRATIC EQUATIONS
1. Quadratic Equation : a
 
x
2
 + b
 
x + c = 0,  a ? 0
x = 
a 2
c a 4 b b
2
? ? ?
, The expression b
2
 ? 4 a
 
c ? D is called
discriminant of quadratic equation.
If ?, ? are the roots, then (a) ? + ? = ? 
a
b
    (b) ? ? ? ? = 
a
c
A quadratic equation whose roots are ? & ?, is (x ? ? ?) (x ? ? ?)
= 0  i.e. x
2
 ? ( ? + ? ) x + ? ? ?  = 0
2. Nature of Roots:
Consider the quadratic equation, a
 
x
2
 + b
 
x + c = 0 having ?
,
 ? as its roots;
D ? b
2
 ? 4 a
 
c
      
    D = 0     D ? 0
Roots are equal ? = ? = ? b/2a Roots are unequal
  
   a, b, c ? R & D > 0 a, b, c ? R & D < 0
    Roots are real       Roots are imaginary ? = p + i
 
q,  ? = p ? i
 
q
     
a, b, c ? Q & a, b, c ? Q &
D is a perfect square   D is not a perfect square
?  Roots are rational ?  Roots are irrational
? i.e. ? = p +
q
, ? = p ?
 
q
a = 1, b, c ? ? & D is a perfect square
? Roots are integral.
Page # 23
3. Common Roots:
Consider two quadratic equations a
1 
x
2
 + b
1 
x + c
1
 = 0 & a
2 
x
2
 + b
2 
x + c
2
 = 0.
(i) If two quadratic equations have both roots common, then
2
1
a
a
 =
2
1
b
b
 =
2
1
c
c
.
(ii) If only one root ? is common, then
? =
1 2 2 1
1 2 2 1
b a b a
a c a c
?
?
 = 
1 2 2 1
1 2 2 1
a c a c
c b c b
?
?
4. Range of Quadratic Expression f (x) = a
 
x
2
 + b
 
x + c.
Range in restricted domain: Given x ? [x
1
, x
2
]
(a) If ?
a 2
b
 ? [x
1
, x
2
] then,
f(x) ? ? ? ? ? ? ? ) ( ) ( ) ( ) (
2 1 2 1
x f , xf max , xf , xf min
(b) If ?
a 2
b
 ? [x
1
, x
2
] then,
f(x) ? 
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
a 4
D
, xf , xf max ,
a 4
D
, xf , xf min ) ( ) ( ) ( ) (
21 21
Page # 24
5.
Let f
 
(x) = ax²
 
+ bx
 
+ c, where a > 0 & a
,
 b
,
 c ? R.
(i) Conditions for both the roots of f
 
(x) = 0 to be greater than a
specified number‘x
0
’ are  b² ? 4ac ? 0; f
 
(x
0
) > 0 & ( ?
 
b/2a) > x
0
.
(ii) Conditions for both the roots of f
 
(x) = 0 to be smaller than a
specified number ‘x
0
’ are  b² ? 4ac ? 0; f
 
(x
0
) > 0 & ( ?
 
b/2a) < x
0
.
(iii) Conditions for both roots of f
 
(x) = 0 to lie on either side of the
number ‘x
0
’ (in other words the number ‘x
0
’ lies between the roots
of f
 
(x) = 0), is f
 
(x
0
) < 0.
(iv) Conditions that both roots of f
 
(x) = 0 to be confined between the
numbers x
1
 and  x
2
, (x
1
 < x
2
) are b²
 
? 4ac ? 0;  f
 
(x
1
) > 0 ; f
 
(x
2
) > 0 &
x
1
 < ( ?
 
b/2a) < x
2
.
(v) Conditions for exactly one root of f
 
(x) = 0 to lie in the interval (x
1
, x
2
)
i.e. x
1
 < x < x
2
 is f
 
(x
1
).
 
f
 
(x
2
) < 0.