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.
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