2.19 U PgtU(Quadratic Equations):

Cd ĺsgv Aizz Pt PƮ v۽P Az tU vUAivۣ. vUz tU CzszjAz Pt tU vAqjvۣ. vUz tU UƮz 4 gjAz Pt PzgU, 6 tUAz , MAzAzjAz Pt gxz Pq, Pt gxz l, v Pt vAqjvۣ. Gz MAz tAz Pt PAzg, v۽P Az Ml J tU vUAivۣ? (ïw: P 71)

ð ĸ rĪ D EzAi?

 

d fêz JzjĪ PɼV ĸU Gvg Uv?

ĸ: ê, ûvg MnU MAz P U Aiĸwj. DgPV Ml 480g. RZUvzAz P Qj. Dzg PƣAi Ptz 8 A ûvg P U. Eg gzzjAz, wAiƧg DgPV 10g. aU PqPv. Uzg wAiƧg PƣU Pl t J?

 

FUU PɼV Pɮ ĸ rĪz Pwzê:

1. MAz Uz vۼv 60lg Dzg Czg Ai Gz J?

P: Uz MAz : x DVg. DU vۼv = 4x

4x =60

x =15 lgU

F jw gSvP PgtP MAz jggvz. F jg Pgtz Ʈ Jvê. E 15, 4x = 60 Jߪ Pgtz Ʈ.

 

2. MAz Uz t 25 Zzg lgUzg Czg MAz Ai Gz J?

P: Uz MAz Ai Gz x DVg. DU D Uz t = x2

x2 = 25 =5*5

x=5 lgU

Dzg 25 = -5*-5 JAz DUvz. x= -5 F Pgtz Ʈ DVz x2 = 25 vȦ rvz. DzjAz x = 5 F Pgtz ƮU.

Uz ī Gz It ASAiiU zs嫮zjAz x = -5 Pgtz jgV vUzPƼŢ.

S: DP zz Wv 2 DVgĪ Pgt U Pgt (quadratic equation) Jvê.

x2 = 25 Ez x2 - 25 =0 JAz gAiĺz (KP v U ?)

F Pgtz ZgPg x Wv 2 ivz. z Wvz ZgPg.(bx JA CA.)

S:

1. ax2 +c = 0 Pgtz gƥz Pۥrzz Pgt z U Pgt (pure quadratic equation)Jߪg. a v c U ۪ ASUVz, a 0 DVgvz.

 

2. a, b v c U ۪ ASUVz a 0, b 0, DVgĪ ax2 +bx+ c = 0 gƥz Pۥrzz Pgt ıU Pgt (Adfected quadratic equation) Jߪg. E b=0 Dzg, Cz z U PgtUvz.

ıU PgtP Gzgu: 3x2 -5x-16=0

 

Gzgu: 3x2 -16=0 F Pgt rĪ.

3x2 =16 (16 sUP vAz, Cx Jgq Pq 16 Prz)

x2 =16/3

x = = /= (4/)

 

2.19 ĸ1: r: x2/2 3/4 = 29/4

 

jg:

PAvg irzU,

x2/2 = 29/4+3/4 = (29+3)/4 = 32/4 =8

x2 =16

x = 4

 

2.19 ĸ 2 : r: (2m-5)2= 81

 

jg:

(2m-5)2= 92

2m-5 = 9

2m = 9 +5 (PAvg irzU)

2m = +9+5 =14 Cx 2m = -9+5 = -4

m= 7 Cx m= -2

v:

m = 7: DzU, (2m-5)2=(9)2=81= .

m = - 2: DzU, (2m-5)2=(-4-5)2=(-4-5)2=(-9)2=81= .

 

2.19 ĸ 3 : c2= a2+b2 DVz a=8, c=17 DzU bAi ɯ K?

 

jg:

c2= a2+b2

b2= c2-a2

b = (c2-a2) (DAzg UƮ)

a v c U ɯU E DzòzU,

b = (c2-a2)

= (172-82)

= (289-64) = (225) = 15

 

v:

a=8, b=15 DzU = a2+b2=64+225 =289 = 172= c2=Jq

 

2.19 ĸ 4 : MAz Aqj wd r Jvg h DzU Czg Ws(Uv) = V = r2h

1. r v K?

2. Uv=176 , Jvg=14DzU Aqj wd PAqĻr.

 

jg:

V =r2h

r2= V/h

r = (V/h)

 

zvA: V=176, h = 14

= 22/7 (åz ɯ)

r2=V/h = 176*7/(22*14)= 4

r = 2

wd It ASAiiUĪŢ. DzjAz r=2 iU.

 

v:

= 22/7, h =14, r=2:

sU= r2h= 22*4*14/7 = 22*4*2=176=V= JqsU

 

2.19.1 ı U Pgt Cv PĢAz rĪz(Solving Adfected Quadratic equations by Factorisation method)

 

F zsz U Pgt Jgq zU UtުV gz, wAiAz ƣU ð, ZgPgz ɯAiģ PAqĻrAivê. F zsP vA CsP v jAiiV PAiĮ vA AiĨP.

 

2.19.1 ĸ 1: r: 6-p2=p

 

jg:

PAvg irzU, zv ĸ: p2+p-6 = 0

FU, JqsU (x+a)(x+b) gƥz gAiĨP.

E a+b =1, ab = -6.

- 6 g CvU AiUU (1, -6), (-1,6), (2,-3), (-2,3), (3,-2), (-3,2)

EU a+b =1, ab = -6. F AiĪP CĸgVgĪ UA

a = -2 and b= 3

p2+p-6 = p2+3p-2p -6

= p(p+3) -2(p+3) ---- i z(p+3) g vUzU

= (p+3)(p-2)

p2+p-6 = 0

(p+3)(p-2) = 0 (Jgq zU Ut 0 Dzg CU MAz z 0 DVgèP.)

p+3 = 0 Cx p-2 = 0

p= -3 Cx p =2 E Pgtz ƮU.

 

v:

p=2 DzU, JqsU: 22+2-6 =4+2-6 = 0 = sU.

p = -3 DzU, JqsU: (-3)2 -3- 6 = 9-3-6 = 0 sU.

 

2.19.1 ĸ 2: r: 6y2+y -15 = 0.

 

jg:

FU JqAiģ (ax+b)(cx+d)={ acx2 + x(ad+bc)+bd} gƥP jwP.

E ac=6, bd= -15, ad+bc =1 DVgP.

jîɬAz a=3, c=2, b=5, d= -3 DVgvz.

6y2+y -15

= 6y2+10y -9y -15

= 2y(3y+5)-3(3y+5) i z3y+5 g vUzU

= (3y+5)(2y-3)

6 y2+y -15 =0 DVgĪzjAz

(3y+5)(2y-3) =0

3y+5 = 0 Cx 2y-3 =0

y = -5/3 Cx y =3/2E Pgtz ƮU.

 

v:

y=3/2 DzU, Jq = 6*9/4 +3/2 -15

=27/2+3/2 -15

= (27+3)/2 15 = 0 =

Ezjw y= -5/3 DzU vɣr.

 

2.19.1 ĸ 3: r: 13m = 6(m2+1)

 

jg:

6m2-13m+6 =0

FU JqsU

(ax+b)(cx+d)={ acx2 + x(ad+bc)+bd} gƥP vgP.

E ac=6, bd= 6, ad+bc = -13 DVgP.

jîɬAz a=3, c=2,b=5,d= -3

i.e 6m2-13m+6=0

= 6m2-9m -4m+6

= 3m(2m -3) -2(2m-3) ------ i z2m-3 gvUzU

= (2m-3)(3m-2)

6m2-13m+6 =0

(2m-3)(3m-2)=0

2m-3 = 0 Cx 3m-2 =0

m = 3/2 Cx m =2/3 E zv Pgtz ƮU.

 

v:

m=2/3 DzU, JqsU = 6*4/9 -13*2/3 +6

= 8/3 -26/3+6

=(8-26)/3 +6 = 0 =sU.

m= 3/2 DzU v r.

 

ð jzAv zsP vA Cs U AiĪǨP.VgĪU KP vAz PAqĻrAiĨgz?

2.19.1 ĸ 4: 2x2+3x+1 =0 Jߪz rĪ.

 

CA.

Av

gu

1

x2 +(3/2)x+ (1/2) =0

Jgq U 2 jAz sVz.

2

x2+(3/2)x= -(1/2)

(1/2) U vAz.

(x+a)2 = x2+2ax+ a2 Jߪ v GAiVĪ Vzݰ Ʈ PAqĻrAiħz. VgĪU 2ax = (3/2)x JAz wAiħz. a =3/4

3

x2+(3/2)x+ (3/4)2 = -(1/2)+ (3/4)2

(3/4)2 Jgq UU Prz.

4

LHS = x2 +2(3/4)x + (3/4)2= [x+(3/4)]2

p2+2pq+q2 = (p+q)2 E p=x, q= 3/4

5

RHS = -(1/2)+ (3/4)2 =-(1/2)+ (9/16)= (1/16)

i bz 4*4=16

6

[x+(3/4)]2=(1/16)

Av 4v 5jAz LHS=RHS

7

(x+(3/4)) = (1/4)

Av 6 g UƮ

8

x = -(3/4) (1/4) = -(1/2) or -1

3/4 UAvj

 

 

ð jzAv zszAv ax2 +bx+ c =0 z Ʈ PAqĻrAiĪ.

 

UPgtz ƮU PAqĻrAiĪ v(Formula for finding roots of the quadratic equation)

 

U Pgtz i gƥ: ax2 +bx+ c =0,E a, b, c U ۫P ASUVz a 0, b 0. F Pgtz ƮU PAqĻrAiĪ.

CA.

Av

gu

1

x2 +(bx/a)+ (c/a) =0

Jgq U aAz sVz.

2

x2 +(bx/a) = -( c/a)

c/a Aiģ U vAz.

3

x2 +(bx/a) + (b/2a)2 = -( c/a) + (b/2a)2

(b/2a)2 Jgq UU Prz.

4

LHS= x2 +(bx/a) + (b/2a)2= [x+(b/2a)]2

p2+2pq+q2 = (p+q)2 E p=x, q= b/2a

5

RHS = b2/4a2- c/a= (b2-4ac)/ 4a2

i bz 4a2

6

[x+(b/2a)]2 =(b2-4ac)/ 4a2

Av 4v 5jAz LHS=RHS

7

x+(b/2a) = ((b2-4ac)/ 4a2)

= ((b2-4ac))/ 2a

Av 6 g UƮ

8

x = [-b (b2-4ac)]/2a

b/2a RHS U PAvj.

 

ax2 +bx+ c =0 Pgtz ƮU:

x = [-b +(b2-4ac)]/2a v x = [-b -(b2-4ac)]/2a

UĤ:

F v Uv Jߪg v Ez xĪV sgwAi Utvd zsgZAi g(1025AD) jZĹgvg. Ez Pgƥz ïwAiĮ Pnz(P 67)

 

2.19.1 ĸ 5: r: 4x2+8x+4 = 0

 

jg:

E FU, a =4, b=8, c =4

b2-4ac = 64 4*4*4 = 0

(b2-4ac) = (0) = 0

 

ƮU: p = [-b +]/2a =(-8+0)/8 = - 1 Cx

p = [-b -]/2a = (-8-0)/8 = - 1

E ƮU ĪV: - 1

 

UĤ: 4x2+8x+4 = 4(x2+2x+1) = 4(x+1)(x+1). F Pg x=-1 Ʈ DVz.

 

2.19.1 ĸ 6: r: p2+p-6 = 0(2.19.1.1 g irz P̪ EzVz.)

 

jg:

Pgt ax2 +bx+ c =0 gƥzz.

a =1, b=1, c =-6

b2-4ac = 1 4*1*(-6) = 25

= (25) = 5

 

vzAv:

p = [-b +]/2a =(-1+5)/2 = 2 Cx

p = [-b -]/2a = (-1-5)/2 = -3

F ƮU F AzAi zgw (2.19.1.1)

2.19.1 ĸ 7: r: 6y2+y -15 = 0 (2.19.1.2g irz)

 

jg:

Pgt ax2 +bx+ c =0 gƥzz.

a=6, b=1, c= -15

b2-4ac = 1 4*6*(-15) = 361

(b2-4ac) = (361) = 19

 

vzAv,

y = [-b +]/2a =(-1+19)/12 = 18/12= 3/2 Cx

y = [-b -]/2a = (-1-19)/12 = -20/12 = -5/3

F ƮU F AzAi qɢzê.

3/2 v -5/3 ƮU DVgĪzjAz (y-3/2)(y+5/3) CvU DV.

(y-3/2)(y+5/3) = (2y-3)(3y+5)/6

6y2+y -15 = (2y-3)(3y+5)

 

ZlĪnP: 2.19.1.3 g Pl ĸAiģ v GAiV r.

 

2.19.1 ĸ 8: r: y2-2y+2 =0

 

jg:

zv Pgt ax2 +bx+ c =0 gƥzz.

a=1, b=-2, c=2

b2-4ac = 4 4*1*2 = -4

(b2-4ac) = (-4) = 2

vzAv,

y = [-b +]/2a =(2+2)/2 = 1+Cx

y = [-b -]/2a = (2-2)/2 = 1-

 

ƮU ۪ ASU.

 

v:

zv Pgtz y= 1+ Dzò,

y2-2y+2 = (1+)2 -2(1+) +2 ((1+)2 (j (a+b)2 =a2+b2+2ab v GAiV)

= [1 +(-1) +2] +[-2 -2] +2

= 1-1 +2-2 -2+2 = 0 = .

Ez jw EAz Ʈ = 1-P vɣr.

 

2.19.1 ĸ 9: r: 2(3y-1)/(4y-3) = 5y/(y+2) -2

 

jg:

= [5y -2(y+2)]/(y+2) = (3y-4)/(y+2)

FU rPzz: 2(3y-1)/(4y-3) = (3y-4)/(y+2)

Cq UuPgAz, 2(3y-1)*(y+2) = (3y-4)*(4y-3)

2(3y2+6y y -2) = 12y2-9y -16y+12

6y2+10y -4 = 12y2-25y +12(PAvjzU)

6y2-35y +16=0

FU F Pgt ax2 +bx+ c =0 gƥzz.

E a=6, b=-35, c= 16

b2-4ac = 1225 4*6*16 = 1225-384 = 841

(b2-4ac) = (841) = 29

vzAv,

y = [-b +]/2a =(35+29)/12 = 16/3 Cx

y = [-b -]/2a = (35-29)/12 = 1/2

 

v

zv Pgtz F ɯU Q v r.

 

2.19.1 ĸ 10: r: (y-1)(5y+6) /(y-3) = (y-4)(5y+6)/(y-2)

 

jg:

Pgtz Cq UuPg irzU,

(y-1)(5y+6)(y-2) = (y-4)(5y+6)(y-3)

JqsU = (5 y2+6y-5y-6)(y-2)

= (5 y2+y-6)(y-2)

= 5 y3+ y2-6y -10 y2-2y+12

=5 y3 -9y2-8y+12

sU = (5y2+6y-20y-24)(y-3)

= (5y2-14y -24)(y-3)

= 5y3-14 y2-24y -15y2+42y+72

= 5y3-29y2+18y+72

JqsU = sU

5 y3 -9y2-8y+12= 5y3-29y2+18y+72. (PAvjzU)

5 y3 -9y2-8y+12-(5y3-29y2+18y+72) =0

20y2-26y-60 = 0 ( 2 gvUzU)

10y2-13y-30 = 0

FU Pgt: ax2 +bx+ c =0 gƥzz.

a=10, b=-13, c= -30

b2-4ac = 169 4*10*(-30) = 169+1200 = 1369

(b2-4ac) = (1369) = 37

vzAv,

y = [-b +]/2a =(13+37)/20 = 50/20 = 5/2 Cx

y = [-b -]/2a = (13-37)/20 = -24/20 = -6/5

 

v:

yAi ɯAiģ Pgtz Dzò LHS = RHS gvz.

ð ĸAiģ rĪ AiiAi zs:-

zv Pgtz (5y+6) i Cv, VU 2 DAiU:- CAzg

(1) 5y+6 = 0:

DU 5y= -6 y = -6/5

y = -6/5 JAz zv ĸAi jg ---------(1)

(2) 5y+6 0 Dzg, 5y+6 jAz Jgq U sV,

[(y-1)/(y-3)] =[(y-4)/(y-2)] :

Cq UuPg ir,

(y-1)(y-2) = (y-4)(y-3)

CAzg y2-2y-y+2 = y2-3y-4y+12

CAzg y2-3y+2 = y2-7y+12: (PAvjzU)

CAzg y2-3y+2-( y2-7y+12)=0

CAzg y2-3y+2-y2+7y-12=0

CAzg 4y-10=0

CAzg 4y=10 or y=10/4 =5/2 ---------------------------(2)

 

(1) v (2) jAz, zv Pgtz iˮU: 5/2 v -6/5

 

2.19.1 ĸ 11: y/(y+1) + (y+1)/y = 25/12

 

jg:

zv Pgtz JqAiģ ĮPjzU,

[y*y +(y+1)(y+1)]/[y(y+1)]

= (y2+y2+2y+1)/( y2+y)

LHS = RHS DVgĪzjAz

(y2+y2+2y+1)/( y2+y) = 25/12

Cq UuPg ir,

12(y2+y2+2y+1) = 25( y2+y)

24y2+24y+12 = 25y2+25y.

JqAiİgĪz U PAqĺV.

0 = y2+y-12

F Pgt ax2 +bx+ c =0 gƥzz.

a=1, b=1, c= -12

b2-4ac = 1 4*1*(-12) = 1+48 = 49

= (49) = 7

vzAv, ƮU:

y = [-b +)]/2a =(-1+7)/2 = 3 Cx

y = [-b -)]/2a = (-1-7)/2 = - 4

 

v:

F ɯU zv Pgtz DzòzU Jq= gvz.

 

2.19.1 ĸ 12 : r: (3x2-5x+2) (3x2-5x-2)=21

 

jg:

1. 3x2-5x = y DVg, DU zv Pgt: (y+2) (y-2) =21

y2 4 = 21

y2 = 21+4 =25

y =(25)= 5

 

2. y = 3x2-5x =5

3x2-5x 5=0

x = - (-5) (25+60)/2*3 = 5 (85)/6

 

2.19.1 ĸ 13 (F sUz DgAsz Pl AS): ê ûvgqUr MAz P U zsjwj. DgPV Ml 480g. RZUvz. Dzg Pƣ Ptz 8 d ûvg P g. UV wAiƧg DgPV 10g. aU PqPv. Uzg PƣAiİ wAiƧg Pl t J?

 

jg:

P U z zsjzg AS: x

DU DgPV wAiƧjU vUĪ Z: 480/x

8 d gzzjAz P zg AS: (x-8)

FU wAiƧjU vUĪ Z = 480/(x-8)

F t AZ zsjz tPV 10g. Z

Ƹzg= zg +10

480/(x-8) = 480/x + 10

480/(x-8) = (480+10x)/x.

Cq UuPg ir,

480x = (480+10x)(x-8)

RHS= 480x -480*8 +10x*x-80x

= 480x - 3840+ 10x2-80x = 10x2+400x-3840

0 =10x2+400x-3840-480x. (PAvjzU)

Cx 10x2-80x-3840 =0

F Pgt 10 jAz sV.

x2-8x-384 =0

F Pgt ax2 +bx+ c =0 gƥzz.

a=10, b= -80, c= -3840

b2-4ac = 6400 4*10*(-3840) = 6400 +153600 =160000

= (160000) = 40

vzAv ƮU:

x = [] =(80+400)/20 = 24 Cx

x = [-b -]/2a = (80-400)/20 = -16

dg ASAi It ASAiiUĪŢ. DzjAz x = 24

24 dg U zsjzg.

PƣAiİ wAiƧg PqPz t (=) = 30g.

v:

24 dg P U zsjzg.

wAiƧg RZ = 480/24 = 20g.

8 d UzzjAz P zg = 24 - 8 = 16

FU wAiƧg RZ = 480/16 = 30g.

Ez zzQAv 10g. Z.

Gvg ĸU vAiiUvz.

 

2.19.1 ĸ 14: MAz APã wPãz Pt 20lg DVz. Gzgq U v 4lg Dzg, Dgq U Gz PAqĻr.

 

jg:

APã wPãz Pt l Gzgq U x, y DVg. xUg AizAv,

(Pt)2 = x2+ y2 . Pt = 20. 202 = x2+ y2 ======= (1)

Ugqg v 4 = x-y = 4: x= 4+y.

x F ɯAiģ Pgt (1)g Dzò, 400 = x2+ y2 =(4+y)2+ y2 = (16+8y+ y2)+ y2

=16+8y+ 2y2. (PAvjzU) 2y2+8y-384 = 0

F Pgt ax2 +bx+ c =0 gƥzz. a=2, b= 8, c= -384.

b2-4ac = 64 4*2*(-384) = 64+3072 =3136 =(3136) = 56

vzAv ƮU: y = [-b +)]/2a =(-8+56)/4 = 12 Cx y = [-b -)]/2a

= (-8-56)/4 = -16

wPãz It ASAiiUĪŢ. y =12. MAz 12,EAz (x=4+y) = 16.

 

v:

()2+ ()2 = 122+ 162 = 144+ 256 = 400 =202 .(Pt)2

ĸ jg jAiiVz.

 

2.19.1 ĸ15: Jgq ltU qī zg 1200Q... MAz gʮUrAi F Jgq ltU qĪ Nqv۪. gʰ U z UQAv 30Q../UA. Zzg CzP Jgq UAm Ai Pr PUvz. Uzg gʰ z U J?

 

jg:

gʰ z U = xDVg.

Nq Pz Ai = 1200/x UA.

U 30 Q../UA. Zzg Jgq Uz Pz Ai = 1200/(x+30).UA.

Ƹ AiĪ Aa AiQAv 2 UAm Pr.

1200/x-1200/(x+30) = 2

ZlĪnP: APæ, v GAiV ƮU PAqĻrj: x=120

 

v:

1200/120 1200/150 = 10-8 =2 zvA.

 

2.19.1 ĸ 16: M P MAz mg ztAiģ Jgq AzgU qĪ Zĸvۣ. AzgU qī zg 8 Q.. C MAz AzjAz gl EAz AzjU V ţB g 1UA 40ĵU P. z U UAmU 2 Q.. Dzg, Ѯ j ztAi U J?

[ztAi z evU ZĪU Ai Pr P. z gz ZĪU Ai eۨP]

 

jg:

ztAi U = xDVg (Ѯ j)

V, g Pz Ml Ai 1UA. 40. = 100/60 = 5/3 UAm.

AzgU qī zg = 8Q..

z U 2Q./UA.

z Q̣ U Pz P = 8/x+2 (ztAi U + z U)

z gz U Pz P = 8/x-2 ( U Pr iqvz)

Ml Ai = 8/(x-2) + 8/(x+2) = 5/3

DzzjAz rPz: 8/(x-2) + 8/(x+2) = 5/3

 

ZlĪnP: Pgt ĮPj, v GAiV Ʈ PAq rj x =10

 

v:

Pz Ml P = 8/(10-2) + 8/(10+2) = 8/8 + 8/12 = 1+2/3 = 5/3 zvA

 

2.19.1 ĸ 17: MAz i Uv AiQAv 30 ĵ vqV gnv. Cz Ait¸Pz zg 1500 Q.. Uv AiP CU vĥ Cz v U iƮ UQAv 250Q.. aѸP. Uzg Czg iƮ U v iƮ C PAqĻrj.

 

jg:

iz vz U = x DVg.

PĸPz zg = 1500Q..

iư Aiitz Ai = zg/ U = 1500/x UAm.

i Czs UAm vqV gnz. Uv üU Uj vĥ Cz v U aѹPƼèP.

FU iP Ait¸ EgĪ Ai = (1500/x) -1/2

Ez Aiz i 1500 Q.. jz. DU U: (x+250)

zg= Ai*Ƹ U

I.e. 1500 = {(1500/x) -1/2}*(x+250) = (3000-2x)*(x+250)/2x

I.e. 3000x = (3000-x)(x+250)

Cq UuPgAz,

I.e. 3000x = 3000x -x2+750000-250x

I.e. x2-750000+250x =0

v GAiV: = 1750

ƮU:

x = [-b )]/2a = (-250 1750)/2

x = 750 Cx x =-1000

iz U It ɯ Cxz x = 750Q./UAm

iư Aiitz Ai= 1500/750 = 2UAm

 

v:

iz U 250Q.. ZzzjAz Ƹ U: 1000Q.. /UAm.

1500Q.. Pĸ Pz Ai = 1500/1000 = 1.5 UAm. CAzg

Uv Aiitz üVAv UAm Pr.

i UAm vqV gngĪzjAz, jAiiz üU UjAiģ vvz. ĸ jg jAiiVz.

 

2.19.1 ĸ 18: K qV, AU UA Ml AU UƮz 7/2 g AU PƼz zqz Dqw۪. Gzgq AU PƼz dUqw۪. Uzg Ml EgĪ AUɵ? (ïw P 70)

 

jg:

Ml AU AS x DVg.

zqz DqwgĪ AU = (7/2)

PƼz dUqĪ AU = 2

*      x= (7/2)+2

F Pgt rzU, ƮU: 1/4 Cx 16

Dzg AU AS 1/4 EgUz. Ml EgĪ AU = 16

v:

16 = 14+2 = (7/2) +2 ĸAiİ PlAvAi Ez.

2.19.1 ĸ 19: Cd ĺsgv Aizz Pt PƮ v۽P Az tU vUAivۣ. vUz tU CzszjAz Pt tU vAqjvۣ. vUz tU UƮz 4 gjAz Pt PzgU, 6 tUAz , MAzAzjAz Pt gxz Pq, Pt gxz l, v Pt vAqjvۣ. Gz MAz tAz Pt PAzg, v۽P Az Ml J tU vUAivۣ? (ïw: P 71)

 

jg:

vUz tU Ml AS x Eg.

Av

KvP

J

1

Pt tU Pvj

(x/2)

2

Pt PzgUUV

4

3

U

6

4

gxz Pq,l,Pt UU

(1+1+1) =3

5

Pt ï

1

 

x = (x/2)+ 4 +6+3+1

x (x/2)-10 = 4

(x/2)-10 = 4

(x-20) = 8

x2-40x+400 = 64x -------- ((a+b)2 v GAiVz).

x2-104x+400 =0

(x-100)*(x-4) =0

x=100 Cx x=4

(C 6 tU U GAiVgĪzjAz 4 zs嫮)

DzzjAz Cd 100 tU vUɢgvۣ

 

v:

100= 50+40+6+3+1

 

2.19.1 ĸ 20:MAz Pr zsz AUU UA 1/5 sUz 3 Pz Vz UA UĺU ìv. Gz MAz gz gAAiģ wv. AUU Ml AS J? (sg : dUtv)

 

jg:

AUU Ml AS x Eg.

Av

J

J

1

UĺU

{(x/5)-3}2

2

Gzz

1

{(x/5)-3}2+1 =x

(x2/25) (6x/5)+9+1=x

(x2/25) (11x/5)+10=0

x255x+250=0

(x-50)*(x-5) =0

x=50 Cx x=5: 5 zs嫮 KPAzg {(x/5)-3} It DUgz

v:

50= (10-3)2+1= 49+1,

 

2.19.2 U Pgtz ƮU s(Nature of roots of a Quadratic equation)

 

ê FgU ĸU rĪU b2-4ac Ai ɯU UĤg?

 

2.19.1 .5g b2-4ac = 0 ƮU gg .

2.19.1 .8g b2-4ac <0 ƮU C۫P ASU.

GzU b2-4ac > 0 ƮU ۫P ASU.

 

 

U PgtU F b2-4acAiģ zs (discriminant) Jvê. Ez () Az avê.

 

U F wêiP gvê.

 

zsPz ɯ (b2-4ac)

=

ƮU s=[-b ]/2a

1

= 0

ƮU ۪ v .

2

>0 (zs AS)

ƮU ۪ASU v ߪVgvz.

3

<0 (It AS)

ƮU Īz C۫P ASU ( ASU)

 

2.19.2 ĸ 1: m Aii zs ɯU mk2-3k+1 =0 Ai ƮU (1) (2) ۪ v (3)C۪ v ?

 

 

jg:

Pgt: mk2-3k+1 =0

E a=m, b= -3, c= 1

b2-4ac = 9 4m

1. ƮU ĪUPzg, b2-4ac =0

(I.e. 9-4m =0, i.e. m = 9/4)

2. ƮU ۪ v ߪUPzg, b2-4ac >0

(I.e. 9-4m >0, i.e. 9 >4m, i.e. m < 9/4)

3. ƮU C۪ v ߪVgPzg,

b2-4ac <0

(I.e. 9-4m <0, i.e. 9 <4m, i.e. m > 9/4)

 

2.19.2 ĸ 2: m Aii ɯU r2-(m+1)r +4 =0 Pgtz ƮU (), (۪ v ), (C۪ v )

 

jg:

Pgt: r2-(m+1)r +4 =0

E a=1, b= -(m+1), c= 4

b2-4ac = (m+1) 2-16

= [(m+1)+4]*[(m+1)-4] ===> (CwzU)

= (m+5)(m-3)

 

1. ƮU ĪUPzg, b2-4ac =0

 

(i.e. (m+5)= 0 Cx (m-3)=0 i.e. m=-5 Cx m=3)

 

2. ƮU ۪ v ߪU, b2-4ac >0

(i.e. (m+5)(m-3) >0) Jgq zU Ut zsVzg Jgq zU zsVgP E Jgq zU ItVgP Jߪz UĤzU,Jgq AzsU zs:

Azs 1: m+5 > 0 v m-3>0

I.e. m> -5 v m>3: Vg m>3 DVg P

Azs 2: m+5 < 0 v m-3<0

I.e. m< -5 v m<3: Vg m<-5 DVg P

 

3. ƮU C۪ v ߪU, b2-4ac <0

(i.e. (m+5)(m-3) <0) Jgq zU Ut ItVzg MAz z zsVz EAz ItVgP Jߪz UĤzU,Jgq AzsU zs:

Azs 1: m+5 < 0 v m-3>0

I.e. m< -5 v m>3: Ez zs嫮

Azs 2: m+5 > 0 v m-3<0

I.e. m< -5 v m<3: Vg m -5 v 3 g zs EgèP.

F jw PAq rgĪz ASgSAi ï PɼU Pt¹zAv Ugwz.

 

 

2.19.2 ĸ 3: (p+1) n2+2(p+3)n +(p+8) =0 F Pgtz ƮU ĪUPzg, pAi ɯ PAqĻr.

 

jg:

zv Pgt ax2 +bx+ c =0 gƥzz.

a=(p+1), b= 2p+6, c= p+8

b2-4ac = (2p+6)2 4*(p+1)(p+8)

= (4p2+24p+36) -4(p2+8p+p+8)

= 4p2+24p+36 -4p2-36p-32

=-12p+4

ƮU ĪUPzg, b2-4ac =0

I.e. -12p+4 = 0

I.e. p=1/3

vzAv, p=1/3 DzU ƮU:

n = [-b ]/2a =[-2(p+3)0) ]/2(p+1) = - (p+3)/(p+1)

= - (10/3)/(4/3) = -5/2

 

v:

n = -5/2 Pgtz Dzò,

(p+1) n2+2(p+3)n +(p+8)

= 25(p+1)/4 -5(p+3) +(p+8)

= 25(p+1)/4 -4p -7

= (25p+25-16p-28)/4

= (9p-3)/4 (p = 1/3 Dzò)

=0/4 = 0 = sU

 

2.19.2 ĸ 4: (3p+1)c2+2(p+1)c+p=0 Pgtz ƮU ĪUPzg. pAi ɯ PAqĻr.

 

jg:

Pgt ax2 +bx+ c =0

E a=(3p+1), b= 2p+2, c= p

b2-4ac = (2p+2)2 4*(3p+1)p

= (4p2+4+8p) -4(3p2+p)

= 4p2+4+8p -12p2-4p

= -8p2+4p+4

= - 4(2p2-p-1)

Pgtz ƮU ĪUPzg, b2-4ac =0

2p2-p-1 = 0

Jq = 2p2-2p+p-1

= 2p(p-1)+(p-1)

= (p-1)(2p+1)

FU 2p2-p-1 = 0 DzjAz (p-1)(2p+1) = 0

p=1 Cx p= -1/2

UĤ: 2p2-p-1 = 0 Ezg ƮU PAqĻrAiĮ Cv zs GAiVzê. vzAv p=1 DzU, ƮU:

c = [-b +]/2a =[-2p-2 0) ]/2(3p+1) = - 4/8 = -1/2

p = -1/2 DzU c U EAz ɯ (Cz) gvz.

UĤ: v Jgq j GAiVAi ð Pgt r pAi ɯ PAqĻrAiħz.

 

 

v:

Pgtz c = -1/2 DzòzU,

(3p+1) c2+2(p+1)c +p

= (3p+1)/4+2(p+1)(-1/2) +p

=(3p+1)/4 (p+1) +p

=(3p+1)/4 -1 (4 i bz ir)

= [(3p+1) -4]/4 (p =1 Dzò)

= 0/4 = 0 = zv Pgtz .

ZlĪnP: p = -1/2 DzU, (3p+1) c2+2(p+1)c +p =0 Pgtz ƮU ĪVAi jQ.

 

2.19.2 ĸ 5: 2y2-py +1 =0 Pgtz ƮU ĪVzg. pAi ɯ PAqĻr.

 

jg:

zv Pgt ax2 +bx+ c =0 gƥzz.

E a=2, b= -p, c= 1

b2-4ac = p2 -8

ƮU ĪVg, b2-4ac =0

p2 = 8 :

p = 2

ZlĪnP: p Ai F ɯAi i ƮU PqvzAz vɣr.

 

2.19.3 U Pgtz ƮUU, CU vUU EgĪ AAzs(Relationship between roots and co-efficients):

 

m v n U UPgt ax2 +bx+ c =0 Ezg ƮUVg.

(x-m)(x-n) = 0

U Pgtz ƮU(m,n) :

x = [-b +]/2a Cx x = [-b -]/2a

m = [-b +]/2a

n = [-b -]/2a

m+n = [-b +]/2a + [-b -]/2a

= -2b/2a = -b/a

mn = [-b +]/2a * [-b -]/2a ( (a+b)(a-b) v GAiVz)

= [ (-b)2- {}2] /4a2

= [b2 -(b2-4ac) ] /4a2

= 4ac/4a2

= c/a

wêi:

1) MAz U Pgtz ƮU v = -b/a

2) MAz U Pgtz ƮU Ut = c/a

 

2.19.3 ĸ 1: x2 +(ab)x+ (a+b) =0 F Pgtz ƮU v v Ut PAqĻr.

 

jg:

zv Pgt ax2 +bx+ c =0 gƥzz.

E a=1, b= ab, c= (a+b)

m+n = -b/a = -ab/1 = -ab

mn =c/a =(a+b)/1 = (a+b)

 

2.19.2 ĸ 2: pr2 = r-5 F Pgtz ƮU v v Ut PAqĻr.

 

jg:

Pgt pr2 r+5= 0

F Pgt a x2 +bx+ c =0 gƥzz.

E a=p, b= -1, c= 5

m+n = -b/a = 1/p

mn =c/a = 5/p

 

2.19.4 zv ƮU AgĪ U Pgt gaĪz (Formation of equation with given roots):

 

m v n U U Pgt ax2 +bx+ c =0 Ezg ƮUVg. DU (x-m)(x-n) = 0

Dzg (x-m)(x-n)

=x(x-n)-m(x-n)

= x2 xn mx +mn

= x2 x(n+m) +mn

= x2 ( m+n)x +mn

U Pgtz i gƥ:-

x2 (ƮU v)x +(ƮU Ut) =0

 

2.19.3 ĸ 1: 2a2-4a+1=0 F Pgtz ƮU p v q Dzg, (p+q)2+4pq v (p3 +q3)U ɯ PAqĻrj v p3 v q3U ƮUUĪAv U Pgt g.

 

jg:

zv Pgt ax2+bx+ c =0 gƥzz.

E a=2, b= -4, c= 1

p+q = -b/a = 4/2 =2

pq =c/a =1/2

(p+q)2+4pq=4+2 = 6

U MAz v Uwz: a3+b3= (a+b) (a2+b2-ab)

p3 +q3

= (p+q)( p2+q2-pq)

= (p+q)[( p2+q2+2pq) -3pq)]

= (p+q)[( p+q)2-3pq]

=2*[4-3/2] = 5 ( (p+q) v pqU ɯ Dzòz)

VU p3 v q3U ƮUVgĪ U Pgt P.

ƮU v = p3 +q3 =5 ( ï PZg irz)

ƮU Ut = p3*q3 = (pq)3 =(1/2)3 =1/8

Pz Pgt:

x2-(ƮU v)x+ (ƮU Ut)= 0

I.e. x2-5x+ 1/8= 0 (8 jAz Ut¹)

8x2-40x+1=0

 

2.19.3 ĸ 2: ƮU p/q v q/p EgĪAv U Pgt ga.

 

jg:

m =p/q, n=q/p

m+n = p/q+q/p = (p2+q2)/pq

mn = p/q*q/p =1

U Pgtz Dz gƥ: x2 (n+m)x +mn= 0

I.e. x2 (p2+q2)x/pq +1 = 0

(pqx2 (p2+q2)x +pq)/pq =0(pq i bz irz)

I.e. pqx2 (p2+q2)x +pq=0

 

2.19.3 ĸ 3: x2+px+q=0 Pgtz MAz Ʈ vAz Ʈz ggzg. 3p2=16q JAz .

 

jg:

zv Pgt ax2+bx+ c =0 gƥzz.

E a=1,b=p,c=q

m v n U Pgtz ƮUVg.

m+n = -b/a = - p mn = c/a = q

MAz Ʈ vAzg 3 gz m =3n DVg.

p = - (m+n) =-(3n+n)= -4n v q =mn=3n*n = 3n2

3p2= 3(-4n)2= 48n2=16*3n2 = 16q(3n2=q)

 

2.19.3 ĸ 4: 4x2-8px+9=0 Pgtz ƮU qī v 4 Dzg p Ai ɯAiģ PAqĻrj.

 

jg:

zv Pgt ax2+bx+ c =0 gƥzz.

E a=4,b=-8p,c=9

m v n U Pgtz ƮUVg.

1) m+n = -b/a = 8p/4 = 2p ===(1)

2) mn= c/a = 9/4 ===(2)

ƮU qī v 4 n = m+4 DVg.

n F ɯAiģ (1)g Dzò.

m+n = 2p

m+m+4 = 2p

2m= 2p-4

m=p-2 ------(3)

n= m+4 Ez (2) g Dzò.

m(m+4) =9/4

m2+4m - 9/4 =0

I.e. (p-2)2+4(p-2) - 9/4 =0 {(3)gAv m=p-2}

p2-4p+4 +4(p-2) - 9/4 =0 {(p-2)2 jzU }

p2-4p+4 +4p-8 - 9/4 =0

p2-4 - 9/4 =0

p2-25/4 =0

p2= 25/4

p = 5/2

 

v:

p =(-5/2) ɯAiģ zv Pgtz Dzò,

4x2-8px+9=0

4x2-8*(-5/2)x+9=0

4x2+20x+9=0

F Pgt ax2+bx+c=0 gƥzz. a=4, b=20, c=9

b2-4ac = 400 4*4*9 = 400-144 =256

= (256) = 16

vzAv,

ƮU: x = [-b +]/2a =(-20+16)/8 = -4/8

x = [-b -]/2a = (-20-16)/8 = -36/8

ƮU qī v 32/8 = 4 Pz Pnz.

 

ZlĪnP: p=5/2 Pq Ez svA gvzAz vɣr.

 

2.19 Pv SAU

 

 

 

AS

ɣqPz CAU

1

ax2 +bx+ c = 0 F U Pgtz ƮU: x = [-b+]/2a v

[-b-]/2a

2

m v nU MAz U Pgtz ƮUzg, (m+n) = -b/a

3

m v n U MAz U Pgtz ƮUzg, (mn) = c/a

4

m v n U MAz U Pgtz ƮUzg, x2 (n+m)x +mn =0