quadraticequations slides 617 4431

7/26/2019 QuadraticEquations Slides 617 4431

1/253

Quadratic EquationMC Sir

1. Introduction, Graphs

2. Inequality3. Theory of Equations : Relation between Roots

1

#olyno$ials

%. Identity

&. Infinite Roots, o$$on Roots

'. (a)i$u$ and (ini$u$ *alues of +uadratic and

Rational unction


2/253

7. General 20 inx andy

8. Condition for General 20 inx andy to be

factorized in two linears


2

9. Location of Roots

10.Modulus Inequalit

11.Lo!arit"# Inequalit


3/253


No. of Questions

3

2008 2009 2010 2011 2012

1 -- 3 2 --


4/253

Quadratic

y = ax2 + bx + c ; a 0

4

a = ea ng coe c en

b = coefficient of linear term

c = absolute term


5/253

y = f(x) = ax2 + bx + c

In case

a = 0 y = bx + c is linear polynomial

a = c = 0 y = bx is odd linear polynomial

5


6/253

Cubic Polynomial

y = ax3 + bx2 + cx + d

a = ea ng coe c en

d = absolute term

6


7/253

Roots of Quadratic Equation

y = ax2 + bx + c

Where = b2

! "ac is called discriminant#

7


8/253

ax2 + bx + c = 0

$um of roots = ! b%a

&roduct of roots = c%a

= b2! " ac

8


9/253

Different Graphs of

Quadratic Expression

9


10/253

Example

Parabola

y

2

y = x2 + 2x + 2 = (x + )2 +

= 22! = ! " * 0

#

x 0 2 3 " ! ! 2 ! 3 ! " ! , - ! -

y 2 , 0 . 2/ 2 , 0 . - -

0x

eading coefficient 1 0

or x = ! y s m n mu -1

10


11/253

In general graph of y = ax2 + bx + c ;

a 1 0 outh facing upard

4

y 1 0 x 5

11


12/253

Example

y = x2! "x + " = (x ! 2)2

= 0

(0, 4)

(2, 0)

Ya > 0

D = 0

X

#

x 0 2 3 " , / ! ! 2 - ! -

y " 0 " 6 / 6 / - -

y 0 x5

eading 7oefficient# 1 0 12


13/253

In general graph of y = ax2 + bx + c ;

a 1 0 outh facing upard

=

y = 0 for only one

value ofx(root)

y 10 x R {root}

13


14/253

Example

y = x2! 3x + 2

= 32! "(2) = 1 0

3

2

a > 0

y#

2

!

y 2 0 0 2 / ! : - -

y 1 0 x (! - )(2 -)

y * 0 x (2)

y = 0 x


15/253

In >eneral

y = ax2 + bx + c

a 1 0 parabola mouth acing upard

#

1 0 ?o distinct real root (parabola

cuts the x axis at 2 distinct point)

15


16/253

Example

y = ! x2

! 2x ! 2 = !(x + )2

!

- 2

- 2 -1x

#

* 0

x 0 2 3 ! ! 2 ! 3 - ! -y ! 2 ! , ! 0 ! . ! ! 2 ! , ! - ! -

eading 7oefficient * 0

y

16


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In >eneral

y = ax2 + bx + c

#

a * 0 mouth facing donard

* 0 no real root

y * 0 x5

17


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Example

y = ! x2 + "x ! "= ! (x ! 2)2

= 0

#

x 0 2 3 " ! - ! -y !" ! 0 ! !" 0 ! - ! -

- 4

21

eading 7oefficient * 0 0x

y

18


19/253

In >eneral

y = ax2

+ bx + c

a * 0 mouth acin donard

#

= 0 (one real root) parabola touch the x axis

y 0 x5

19


20/253

Example

y = ! x

2

+ 3x ! 2 = ! (x ! )(x ! 2)D > 0

#

! - ! -y ! 2 0 0 !2 !6 !6 !2 ! - ! -

210

-2

Leading Coefficient < 0

@

A

20


21/253

In >eneral

y = ax2 + bx + c

a * 0 &arabola mouth facing donard

#

o s nc rea roo ara o a cuthe xBaxis at to distinct points#

21


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Co-ordinate of vertex

y = ax2 + bx + c

x =

y =

22


23/253

Natue f Rts

1 0 roots are real C distinct (uneDual)

= 0 roots are real C coincident (eDual)

* 0roots are imaginary#

23


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Natue f Rts

7onsider the Duadratic eDuation ax2 + bx + c = 0

here a b c C a0 then;

If is a perfect sDuare then roots are rational#

24


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Nte

If E = p + is one root in this case (here p is

rational C is a surd) then other root ill be

p B

25


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Nte

If p + iD is one root of a Duadratic eDuation

then the other root must be the conFugate

p ! iD C Gice Gersa# (p D5 C i = )#

2


27/253

Example

et a 1 0 b 1 0 and c 1 0# ?hen both the roots

of the eDuation ax2 + bx + c = 0

#

(a)are real and negatiGe

(b) haGe negatiGe real parts

(c) haGe positiGe real parts

(d) Hone of the aboGe II?BJKK 6.6L

27


28/253

Example

Moth the roots of the eDuation

(x ! b) (x ! c) + (x ! a) (x ! c) + (x ! a) (x ! b) = 0

#

are alays

(a) positiGe (b) negatiGe

(c ) real (d) Hone of these

II?BJKK 60L

28


29/253

Example

?he number of real solutions of the eDuation

NxN2 B 3 NxN + 2 = 0 is

#

(a)" (b)

(c) 3 (d) 2

II?BJKK 62L29


30/253

Example

et f(x) be a Duadratic expression hich is

positiGe for all real Galues of x#

#

If g(x) = (x) + (x) + (x) then or any real x

(a)g(x) * 0 (b) g(x) 1 0

(c) g(x) = 0 (d) g(x) 0

II?BJKK 660L

30


31/253

etbe the roots of the eDuation

(x ! a) (x ! b) = c c0

Example

#

?hen the roots of the eDuation

(x !) (x !) + c = 0 are

(a)a c (b) b c (c) a b (d) a + c b + c

II?BJKK 662L

31


32/253

?rue % 'alse

If a * b * c * d then the roots of the eDuation

Example

#

(x ! a) (x c) + 2 (x ! b) (x d) = 0

are real and distinct# II?BJKK 6"L

32


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?he number of points of intersection of to

Example

#

curGes = 2 in and = , + 2x + 3 is

(a)0 (b) (c) 2 (d) -

II?BJKK 66"L

33


34/253

Example

'or all x x2 + 2ax + 0 ! 3a 1 0

then the interGal in hich a lies is

#

(a)a *, (b) ! , * a * 2

(c) a 1 , (d) 2 * a * ,

II? ! JKK 200"L

34


35/253

Example

If b 1 a then the eDuation (x ! a) (x ! b) ! = 0

has

#

(b) both roots in (B- a)

(c) both roots in (b +-)

(d) one root in (B- a) and the other in (b +-)

II?BJKK 2000L 3


36/253

36


37/253

If the eDuation

sin"

x(O + 2) sin2

x(O + 3) = 0

#

(9) ("2) (M) 3 2)

(7) ("3) () 32L

37


38/253


39/253

[Multiple Objective Type]

?he graph of the Duadratic polynomial;

= ax2 + bx + c is as hon in the i ure # ?hen

#3

(9) b2"ac 1 0 (M) b * 0

(7) a 1 0 () c * 0

y

x

39


40/253

If a b c 5 such that a + b + c = 0 and a c

then proGe that the roots of

(b + c ! a) x2 + (c + a ! b) x + (a + b ! c) are

#"

0


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'ind the Galue of a for hich the roots of the

eDuation (2a ! ,) x2 ! 2 (a ! ) x + 3 = 0 are

eDual#

#,

1


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'or hat Galues of m does the eDuation

x2! x + m = 0 possess no real roots R

#/

2


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'or hat Galues of m does the eDuationx2! x + m2 = 0 possess no real roots R

#.

3

R l ti b t t d


44/253

Relation between root and

Coefficient of Quadratic Equation

ax2 + bx + c = 0 ; a 0 abc5

ax2 + bx + c = a (x ! E) (x !) = 0

+ = C =

44

F ti f


45/253

Formation of

Quadratic Equation

x2! (sum of roots) x + product of roots = 0

5

E l


46/253

Eapl

'orm a uadratic KDuation ith rational

coefficients hose one root is tan.,S

#

6

E l


47/253

Eapl


coefficients hose one root is cos3/S

#

4

E l


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Eapl


coefficients hose one root is tanT%

#

4

Inequalities


49/253

Inequalities

5ules P

9dding positiGe number bothBsides ineDuality

remains same#

Kxample P

2 1 3 1 2

9

Inequalities


50/253

Inequalities

5ules P

$ubtracting both sides by positiGe number

ineDuality remains same

Kxample P

2 1 1 0

50

Inequalities


51/253

Inequalities

5ules P

ultiply C diGide by positiGe number ithout

affecting ineDuality

Kxample P

" 1 2 1 U

51

Inequalities


52/253

Inequalities

5ules P

ultiply C diGide by negatiGe number to

change sign of ineDuality

Kxample P

2 1 ! 2 * !

52

Type 1


53/253

Kxample P

Type 1

Kxpression hich can not be

factoriQed

x2 + x + 1 0

53

Type 1


54/253

Kxample P

Type 1


factoriQed

x2! 3x + " * 0

5

Type 1


55/253

Kxample P

Type 1


factoriQed

3x2! .x + / 1 0

55

Type 1


56/253

Kxample P

Type 1


factoriQed

! x2! 2x ! " 1 0

56

Type 2


57/253

5ules P

Type 2

Kxpression hich can be

factoriQed

aOe coefficient of x as in all linear by

multiplying diGiding by appropriate numberarO Qeros of linear on number line

>iGe sign to respectiGe area on number line57

Type 2


58/253

Type 2


factoriQedx x x

58

Type 2


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Type 2


factoriQedx x

59

Type 2


60/253

Type 2


factoriQedx x ! x x

60

Type 3


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(x2! ,x + /) (x2! /x + ,)0

Type 3

61

Type 3


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2 ! x ! x2 0

Type 3

62

Type 3


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3x2! .x + " 0

Type 3

63

Type 4


64/253

5ules P

Type 4

5epeated inear 'actor

>et rid of eGen poer

odd poer treat as linear

6

Type 4


65/253

yp


(x + (x 3 (x 2 2 1 0

65

Type 4


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yp


(x + / (x + 2 2 (x 3 1 0

66

Type 4


67/253

(x ! 2 (x + 3 (x ! " * 0

yp


67

Type 5


68/253

yp

5ational IneDuality

68

Type 5


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yp

5ational IneDuality

69

Type 5


70/253

yp

5ational IneDuality

70

Type 5


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yp

5ational IneDuality

71

Type 5


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5ational IneDuality

72

Type 5


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5ational IneDuality

Type 5


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5ational IneDuality

4

Example


75/253

#

75

Example


76/253

#

76

Example


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#

Example


78/253

et y =

'ind all the real Galues of x for hich y taOes

#

real Galues# II?BJKK 60L

Example


79/253

'ind the set of all x for hich

II?BJKK 6.L

#

79

Example


80/253

$olGe Nx2 + "x + 3N + 2x + , = 0 II?BJKK 6L#

80

Example


81/253

et a and b be the roots of the eDuation

x2! 0cx ! d = 0 and those of

x2 ! 0ax ! b = 0 are c d# ?hen find the

#

Galue of a + b + c + d hen abcd#

II?BJKK 200/L

81

Example


82/253

etbe the roots of the eDuation

x2! px + r = 0 and%2 2

be the roots of the eDuation x2! Dx + r = 0#

#

?hen the Galue of r is

(a)2%6 (p ! D) (2D ! p) (b) 2%6 (D ! p) (2p ! D)

(c) 2%6 (D ! 2p) (2D ! p) (d) 2%6 (2p ! D) (2D ! p)

II?BJKK 200.L

82

Example


83/253

'ill in the blanO P

If 2 + i is a root of the eDuationx2 + px + D = 0

#

here p and D are real then (p D) = (VV)#

II?!JKK 62 L

Example


84/253

'ill in the blanO P

If the products of the roots of the eDuation

x2! 3Ox + 2e2 log O! = 0 is .

#

then the roots are real for O = V## #

II?BJKK 6"L

4

Example


85/253

If x y and Q are real and different and

u = x2 + "y2 + 6Q2! /yQ B 3Qx ! 2xy then u

#

(a)nonBnegatiGe (b) Qero

(c) nonBpositiGe (d) none of theseII?BJKK 6.6L

85

Example


86/253

If one root is sDuare of the other root of the

eDuation x2

+ px + D = 0 then the relation

beteen and is

#

(a)p3! (3p ! ) D + D2 = 0

(b) p3! D(3p + ) + D2 = 0

(c) p3 + D(3p ! ) + D2 = 0

(d) p3

+ D(3p + ) + D2

= 0 II?BJKK 200"L86


87/253

?he sum of all the Galue of m for hich the#


88/253

roots x and x2 of the Duadratic eDuation

x2! 2mx + m = 0 satisfy the condition

(9) (M)

(7) ()

If andare the roots of the eDuation#2


89/253

ax2 + bx + c = 0 then the sum of the roots of

the eDuation2 2 2! 2! =

in terms ofandis giGen by

(9)! (2!2) (M) (+)2! 2

(7)2+2! " () ! (2 +2)

89

?he set of Galues of a for hich the#3


90/253

ineDuality (x B 3a) (x B a B 3) * 0 is satisfied

for all x 3L is P

(7) (B 2 0) () (B 2 3)

90

Ifandare the roots of a(x2! ) + 2bx = 0#"


91/253

then hich one of the folloing are the roots

of the same eDuationR

!

(7) ()

91

$olGe the folloing IneDuality#,


92/253

92



93/253

93



94/253

(x ! ) (3 ! x) (x ! 2)2 1 0

9



95/253

95


96/253

Double Inequality

96

Example


97/253

$olGe the folloing IneDuality#

97

Example


98/253


98

Example


99/253


99

Example


100/253


G

100

Example

?rue % 'alse P


101/253

?rue % 'alse P

y = ax2 + bx + c@

#

A

101

Example

?rue % 'alse P


102/253

?rue % 'alse P

y = ax2 + bx + c@

#

A

102

Example

?rue % 'alse P


103/253

?rue % 'alse P

y = ax2 + bx + c@

#

A

103

Example

?rue % 'alse P


104/253

?rue % 'alse P

y = ax2 + bx + c@

#

A

10

Example

?rue % 'alse P


105/253

ue % alse P

y = ax2 + bx + c@

#

A

105

Example

?rue % 'alse P


106/253

y = ax2 + bx + c@

#

A

106

Example

?rue % 'alse P


107/253

y = ax2 + bx + c@

#

A

107

Example


108/253

# uadratic KDuation ax2 + bx + c = 0 has no

real roots then sho that c (a + b + c) 1 0

108

Example


109/253

# 'ind a

(a ! ) x2! (a + ) x + a + 1 0 x5

109

Example


110/253

# 'ind a if (a + ") x2 ! 2a x + 2a ! / * 0

x5

110

Example


111/253

# If is root of x2! 2x + , = 0

'ind the Galue of3 +2 B+ 2

111

Example


112/253

# If is root of x2! 2x + , = 0

'ind the Galue of3 + "2 B .+ 3.

112

Example


113/253

# If x = 3 +

'ind the Galue of x" + 2x3 + ""x2! "x + .

113

Example


114/253

# If p (D ! r) x2 + D (r ! p) x + r (p ! D) = 0 has

eDual root#

$ho that P

11

Example


115/253

# If x2 + = " ; x 1 0 then (7)

(a) x3 + xB3 = /2 (b) x 3 + xB3 = ,2

(c) x, + xB, = /2" (d) x , + xB, = .2"

115

Example


116/253

# 'ind the integral solutions of the folloing

system of ineDualities

(a) ,x ! * (x + )2 * .x ! 3

(b)

II?BJKK 6.L

116

Example


117/253

# If l m n are real lm then the roots of the

eDuation (l ! m) x2 ! , (l + m) x ! 2(l ! m) = 0

are

(a) real and eDual (b) complex

(c) real and uneDual (d) none of theseII?BJKK 6.6L

117

Example


118/253

# 'or hat Galue of m does the system of

eDuations 3x + my = m 2x ! ,y = 20 has

solution satisfying the conditions x 1 0 y 1 0

II?BJKK 60L

118

Example


119/253

# 'ind all real Galues of hich satisfy

x2! 3x + 2 1 0 and x2! 3x ! "0#

II?BJKK 63L

119

Example


120/253

# et a b c be real numbers ith a 0 and let

be the roots of the eDuations

ax2 + bx + c = 0#

Kxpress the roots of a3x2 + abc x + c3 = 0

in terms of# II?BJKK 200L

120

Example

2


121/253

# If andare the roots of x2 + px + D = 0 and

are the roots of x2 + rx + s = 0 then

eGaluate (B ) (B ) (B ) (B ) in terms

of p D r and s#

II?BJKK 6.6L

121


122/253

122

$olGe the folloing ineDualities


123/253

#

123

2



124/253

#2

12

3



125/253

#3

125

"



126/253

#"

126

,



127/253

#,

127

/



128/253

#/

128

#. 'or hat Galues of c does the eDuation



129/253

f D

(c ! 2) x2

+ 2 (c ! 2) x + 2 = 0

129

# 'or hat Galues of a does the eDuation



130/253

# 'or hat Galues of a does the eDuation

130

possess eDua roo s

6 'ind the Galue of O for hich the curGe



131/253

#6 'ind the Galue of O for hich the curGe

y = x2 + Ox + " touches the 8x axis#

131

#0 'ind the least integral Galue of O for hich



132/253

the eDuation x2

! 2 (O + 2) x + 2 + O2

= 0 has

132

#

# If the eDuation "x2 ! "(,x + ) + p2 = 0 has



133/253

one root eDuals to to more than the other

133

(a) (b),

(c) , or B (d) " or B3

#2 &ossible Galues of x simultaneously satisfying



134/253

the system of ineDualities

13

(9) (B 3L/ -) (M) (B2 3L/ -)

(7) (B2 B) (" -) () 3 /L

Identity

2


135/253

ax2 + bx + c = 0

Humber of roots are infinite

When a = b = c = 0

135

Note


136/253

3 distinct real root of Duadratic infinite root

136

Example

'ind the Galue of p for hich the eDuation#


137/253

f p f D

(p + 2) (p ! ) x2 + (p ! ) (2p + ) x + p2! = 0

has infinite roots

137

#

Example


138/253

138

Quadratic With One Root Zero

2 + b + 0


139/253

ax2 + bx + c = 0

&roduct of root = = 0

c = 0

139

Quadratic With Both Root Zero

2 + b + 0


140/253

ax2 + bx + c = 0

$um of root = &roduct of root = 0

b = 0 c = 0

10

Quadratic With One Root Infinite

2 + b + 0


141/253

ax2 + bx + c = 0

a = 0

11

Quadratic With Both Root

y = ax2 + bx + c


142/253

y = ax2 + bx + c

a = 0 b = 0 c 0

12

Example

2


143/253

# If (2p ! D) x

2

+ (p ! ) x + , = 0 has both

13

Symmetric Function


144/253

If f() = f()

en s ca e ymme r c unc on o

1

Example

# 7hecO if f() is symmetric or not


145/253

(i) f() =2

+2

= cos B

(iii) f () = sin (B)

(iv) f () = (2 B)

15


146/253

16

Condition for both Root Common


147/253

ax2 + b

x2 + c

= 0

a2x2 + b

2x2 + c

2= 0

17

Condition for !ne Root Common


148/253

18

Example

# 'ind O for hich eDuations x2! 3x + 2 = 0


149/253

3x2

+ "Ox + 2 = 0 haGe a common root

19

Example

# 'ind p and D if px2 + ,x + 2 = 0


150/253

3x2

+ 0x + D = 0 haGe both roots in common

150

Example

# 'ind the Galue of a C b if x2! "x + , = 0


151/253

x2

+ ax + b = 0 haGe a common root here a

151

Example

# If "x 2sin2! ("sin) x + = 0 C

2 2 2 2 2 2 2 2 2 2 2


152/253

a2

(b2

! c2

) x2

+ b2

(c2

! a2

) x2

+ c2

(a2

! b2

) = 0

aGe a common roo an e econ

eDuation has eDual roots find possible Galue

of here(0)

152

Example

# If the Duadratic eDuation

2 2


153/253

ax2

+ bx + c = 0 C x2

+ cx + b = 0

c aGe a common roo en proGe

that there uncommon roots are roots of

the eDuation x2 + x + bc = 0

153

Example

# x2 + ax + 2 = 0 x2 + bx + , = 0 C

2


154/253

x2

+ (a + b) x + 3/ = 0

aGe a common pos Ge roo

'ind a b C common root of eDuation#

15

Example

# If one root of Duadratic eDuation

2


155/253

x2

! x + 3a = 0 is double the root of the

eDua on a = n a

155

Example

# If (x) = x2 + (O ! 26) x ! O

2


156/253

2(x) = 2x2

+ (2O ! "3) x + O

o are ac ors o a cu c po ynom a n

156

Example

# If x 2 + abx + c = 0 C x2 + acx + b = 0


157/253

haGe only one root common then sho

a Dua ra c eDua on con a n ng e r

other common roots is

a(b + c) x2 + (b + c) x ! abc = 0

157

Example

# 9 Galue of b for hich the eDuations

2 2


158/253

x2

+ bx ! = 0 x2

+ x + b = 0

aGe one roo n common s

(a) (b) (c) (d)

II?BJKK 20L

158

Example

'ill in the blanO P

# If the Duadratic eDuations x2 + ax + b = 0


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and x2 + bx + a = 0 (a b) haGe a

of a + b is V #

II?BJKK 6/L

159


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"i#mt $

160

# 'ind Galue of O for hich the eDuation

(x ! ) (x ! 2) = 0 C 2x2 + Ox ! = 0

haGe a common root


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161

#2 If x be the real number such that x3 + "x + #

then the Galue of the expression x.+ /"x2 is

(9) 2" (M) 2, (7) 2 () 32


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162

#3 If eGery solution of the eDuation

3 cos2x ! cosx ! = 0 is a solution of the

eDuation a cos22x + bcos2x ! = 0# ?hen theGalue of (a + b) is eDual to


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(9) , (M) 6 (7) 3 () "

163

#" If x2 + 3x + , = 0 C ax2 + bx + c = 0 haGe

common root%roots and a P b c H then find

minimum Galue of a + b + c


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16

#, etermine the Galues of m for hich the

eDuation 3x2 + "mx + 2 = 0 and 2x2 + 3x ! 2

may haGe a common root#


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165

#/ #. 'or hat Galue of a is the difference

beteen the roots of the eDuation (a ! 2) x2!

(a ! ") x ! 2 = 0 eDual to 3 R


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166

#. 'ind all Galues of a for hich the sum of the

roots of the eDuation x2! 2a (x ! ) B = 0 is

eDual to the sum of the sDuares of its roots#


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167

# 'or hat Galues of a do the eDuations

x2 + ax + = 0 and x2 + x + a = 0

haGe a root in common R


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168

Maximum & Minimum

Value of Quadratic Equation2


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y = ax + bx + c attain its maximum orm n mu a po n ere =

according as a * 0 or a 1 0#

aximum and inimum Galue can be

obtained by maOing a perfect sDuare#

169

Example

p(x) = ax2 + bx + is Duadratic polynomial##


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inimum Galue of p(x) is / hen x = 2

170

Example

y = 2x2! 3x + find minimum Galue of y#


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171

Example

y = . + ,x ! 2x2find maximum Galue of y#


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172

Example

'or x2 smallest possible Galue of

l 0 (

3

"

2

+ + 2/) l 0 ( + 2)

#


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log0(x! "x + x + 2/) ! log0(x + 2)

173

Range of Linear

y = ax + b ;a 0


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17

# y = f(x) = x +

Example


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175

Range of


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y =

y5 !

176

#

Example

'ind range of y


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177

#

Example

'ind range of y


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178

#

Example

'ind range of y


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179

#

Example

'ind range of y


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180

Range of

9ssume y


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9ssume yec or common roo s n numera or

denominator

'orm uadratic KDuation

9pply 0 (since x is real)

$olGe ineDuality in y and hence the range 181

No

9lays checO for coefficient of x2 not eDual


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9lays checO for coefficient of x2 not eDual

to Qero

182

Example

'ind range of folloing


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#

183

Example



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#

18

Example


#


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#

185

Example


#


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#

186

Example


#


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#

187

Example


#


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#

188


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189

# 'ind the range of the function f(x) = x2! 2x ! "


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190

#2 'ind the least Galue of


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191

#3 'ind 5ange


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192

#" 'ind the domain and 5ange of


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193

General 2 in x & y

f(x y) = ax2 + 2h xy + by2 + 2gx + 2fy + c


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f( y) y y g fy

19

Condition of General 2 inx&y

to be Resolved into two linearFactors

2 2 2


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2 2 2 ! !

195

Rule

$tep P

factoriQe purely 2S


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9dd constant to both the linear$tep 3 P

7ompare coefficient of x C coefficient of y Cabsolute term if needed

196

Example

# &roGe that the Kxpression

2x2 + 3xy + y2 + 2y + 3x +

can be factoriQed into to linear factors C


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find them

197

Example

# &roGe that the Kxpression

x2! 3xy + 2y2! 2x ! 3y ! 3, = 0

can be factoriQed into to linear factors C


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find them

198

Example

# If the eDuation x2

+ /y2

! 3x + 2 = 0 is

satisfied by real Galues of x C y then sho

that x 2L C yB% %L


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199

Theory of Euation

ax2 + bx + c = a (x B) (x B)


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ax3 + bx2 + cx + d = a(x B) (x B) (x B)

200

Sum & Product of Root

taken 1 at a time

+ + = Bb%a


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= B a

201

Sum of root taken 2 at a time

+ + = c%a


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202

Bi Quadratic

ax" + bx3 + cx2 + dx + e = a(x B) VV# (x !)


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203


+ + + + + = c%a


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20


+ + + = Bd%a


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205

Note

(a + b + c)2 = a2 + 2ab


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206

me

# 'ind sum of sDuare C sum of cubes of roots

of the cubic eDuation x3! px2 + Dx ! r = 0


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207

me

# $olGe the cubic

"x3 + /x2! 6x ! 3/ = 0

Where su o 2 root is ero


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208

me

# If a b c are roots of cubic x3! x2 + = 0

'ind


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209

me

# If are roots of the eDuation

tan = 3 tan3x

'ind the Galue o tan+ tan + tan + tan


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210

me

# 'ind a cubic each of its roots is greater by

unity then a root of x3! ,x2 + /x ! 3 = 0


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211

me

# 'ind the cubic hose roots are cubes of the

roots of x3 + 3x2 + 2 = 0


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212

me

# ?he length of side of a are roots of the

eDuation x3! 2x2 + ".x ! /0 = 0

'ind2


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213


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21

Tpe !1

Moth roots of a Duadratic eDuation are greater

than a specified number

( ) 1 d


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() 1 d

215

"ondition

If y = ax2 + bx + c

(i) a 1 0

d


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(iii)

(iG) f(d) 1 0

216

d

me

# 'ind the Galue of d for hich both roots ofthe eDuation x2! /dx + 2 ! 2d + 6d2 = 0 are

greater than 3


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217

me

# 'ind all the Galues of Xa4 for hich bothroots of the eDuation x2 + x + a = 0 exceed

the Duantity Xa4#


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218

Tpe ! 2

Moth roots lies on either side of a fixed number

say (d)* d *


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219

"ondition

a 1 0

f(d) * 0 d


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220

# 'ind O for hich root of the eDuation is

greater than 2 and other is less than 2

2 O + + O 2 + O = 0

me


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221

# 'ind the set of Galue of Xa4 for hich Qeroes

of the Duadratic polynomial

a2 + a + 2 + a ! x + a2 are located on

me


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either side of 3#

222

# 'ind a for hich one root is positiGe one is

negatiGe !x2! (3a ! 2) x + a2 + = 0

me


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223

# 'ind a for hich both root lie on either side

of B of Duadratic

a2! ,a + / 2 a 3 + . = 0

me


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22

Tpe ! 3

Moth roots lies beteen to fixed number

d * * * e


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225

"onditions

(i) 0

(ii) f (e) 1 0

iii d 1 0 e

(i ) d


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(iG) d * * e

226

me

If(B/ )

'ind O or hich

Q.


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x2

+ 2 (O ! 3) x + 6 = 0

227

Tpe ! #

Moth roots lies on either side of to fixed number

* d * e *


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228

"onditions

(i) f (d) * 0

(ii) f (e) * 0d e


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229

me

# 'ind O for hich one root of the eDuation

(O ! ,) x2 ! 2Ox + O ! " = 0 is smaller


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230

Tpe ! $

Kxactly one root lies in the interGal (d e)

f(d) f(e) * 0 d


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231

f( ) f( )

e

me

# 'ind the set of Galues of m for hich exactly

one root of the eDuation

2 2 !


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232

me

# 'ind a for hich exactly one root of the

Duadratic eDuation x2 ! (a + ) x + 2a = 0


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233

Type - 6

If f (p) f(D) * 0

Kxactly one root Dp

lies beteen (p D)


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23

Example

# If a * b * c * d sho that

uadratic (x ! a) (x ! c) +(x ! b) (x ! d) = 0


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235

Example

# 'ind p for hich the expression

x2 ! 2px + 3p + " * 0 is satisfied for at least


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236

Example

# 'ind a for hich expression

(a2 + 3) x2 + * 0 is satisfied for at


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237

Example

# 'ind m if x2! "x + 3m + 1 0 is satisfied for

all positiGe x


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238

Example

# $ho that for any real Galue of a

(a2 + 3) x2 + (a + 2) x ! , * 0 is true for at least

#


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239

Example

# If f(x) = "x2 + ax + (a ! 3) is negatiGe for at

least one negatiGe x find all Galues of a


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20

Example

# 'ind a for hich x2 + 2(a ! ) x + a + , = 0

has at least one positiGe root#


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21

Example

# 'ind p for hich the least Galue of

"x2! "px + b2! 2p + 2 in x02L is eDual to 3


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22

Example

# 'ind O for hich the eDuation

x" + x2 ( ! 2O) + O2! = has


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23

Example


x" + x2 ( ! 2O) + O2! = has


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2


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Example


x" + x2 ( ! 2O) + O2! = has


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26

Example


x" + x2 ( ! 2O) + O2! = has


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27

Modulas Inequality


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28

Example

#


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29

Note

| x | < x (-, )

| x | > x (-, -) (,)


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250

Example

Q. (| x 1 | 3) (| x + 2 | 5) < 0


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251

Example

Q. | x 5| > | x2 5x + 9 |


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252

Example

Q.


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253

quadraticequations slides 617 4431

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