4.7
ju{ ;dLs/0f (Quadratic Equations):
gd'gf 1 :
b'O{ cª\sx¿sf] ;ª\Vof df cª\sx¿sf] u'0fg kmn 18 / of]ukmn 9 5 eg] Tof] ;ª\Vof kTtf
nufpg'xf];\ . (If two digits number, the product of the digits is 18 and their sum is 9. Find the
number.)
;dfwfg
b'O{ cª\sx¿ x / y dfgf}F
b'O{ cª\sn] ag]sf] ;ª\Vof= 10x+y
ta, k|Zgsf] klxnf] ;t{cg';f/,
x ×y = 18
y=
_________ (1)
k|Zgsf] bf];|f] ;t{cg';f/,
x+y = 9
18
or, x+
=9
[;dLs/0f (1) af6]
x
x2+18
or,
=9
x
2
or, x +18 = 9x
or, x2-9x+18 = 0
or, x2-6x-3x+18 =0
or, x(x-6) -3 (x-6) =0
or, (x-6)-3(x-6) = 0
or, (x-6) (x-3) = 0
Either,
x-6 =0
 x= 6
or, x-3 = 0
x=3
x sf] dfg ;dLs/0f (1) df /fVbf,
18
x=6 x'“bf, y=
=3
6
x=6/y=3
pSt
;ª\Vof= 10x+y = 10×6+3 = 63
k'g, x= 3 x'“bf, y =18
=6
3
x=3/ y=6
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
166
;ª\Vof=10x +y = 10×3+6 = 36
pSt ;ª\Vof= 63 jf 36 /x]5 .
pSt
1. cEof;sf
(a)
nflu k|Zgx¿
b'O{ ;ª\Vofsf] of]u kmn 7 / u'0fg kmn 12 5 eg] tL ;ª\Vofx¿ kTtf nufpg'xf];\ .
b'O{ cª\ssf] Pp6f ;ª\Vof To;sf cª\sx¿sf] of]usf] rf/ u'0ff 5 . olb
cª\sx¿sf] u'0fg kmn 8 eP Tof] ;ª\Vof kTtf nufpg'xf];\ .
b'O{ cf]6f wgfTds ;ª\Vofx¿dWo] 7'nf] ;ª\Vof ;fgf] ;ª\Vofsf] bf]Aa/eGbf 2 n] a9L 5 /
ltgLx¿sf] u'0fg kmn 12 5 eg] tL ;ª\Vofx¿ kTtf nufpg'xf];\ .
(b)
(c)
pTt/x¿
(a) 3, 4
gd'gf
(b) 24
(c) 2, 6
2:
b'O{ cf]6f ;ª\Vofx¿sf] of]u kmn
lgsfNg'x]f;\ .
;dfwfg
b'O{ cf]6f ;ª\Vof¿ j|mdzM x /
ta, k|Zgsf] klxnf] ;t{cg';f/,
y
21
5 / tL ;ª\Vofx¿sf] ju{sf] of]u
261
5 eg] tL ;ª\Vofx¿
dfgf}F,
x +y = 21
or, y = 21- x _________(1)
k|Zgsf] bf];|f] ;t{cg';f/,
or,
+
or,
or,
or, 2
or, 2(
or,
or,
= 261,
+(21- x)2 = 261
+(21)2-2×21 ×
+441-42x +
x+
= 261
-261 = 0
-42x +180 = 0
-21x +90) = 0
-21x +90 = 0
-15x -6x +90 = 0
or, x (x -15)-6(x -15) = 0
or, (x -15_) (x -6) = 0
Either,
x -15 = 0
 x = 15
x = 15 /fVbf,
y = 21-15 = 6
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
167
 x = 15 / y = 6
or, x -6 = 0
x=6
x sf] dfg ;dLs/0f (1) df /fVbf,
x = 6 /fVbf, y = 21-6 = 15
 x = 6 / y = 15
pSt
b'O{ ;ª\Vof¿ j|mdzM
2. cEof;sf
(a)
nflu k|Zgx¿
b'O{ ;ª\Vofx¿sf] of]u
nufpg'xf];\ .
15
/ 6 jf
16
6
/
15
/x]5g\ .
5 / ltgLx¿sf] ju{sf] hf]8
130
5 eg] tL ;ªVofx¿ kTtf
The sum of two numbers is 16 and the sum of their sTtuares is 130. Find the numbers. [059A2]
(b) b'O{ cf]6f j|mdfut hf]/ ;ª\Vofx¿ kTtf nufpg'xf];\ h;sf ju{x¿sf] of]u kmn 340 5 .
Find two consecutive even numbers of which the sTtuares have the sum 340.
(c) b'O{ cf]6f nuftf/ cfpg] wgfTds k"0ff{ªs
\ x¿sf] u'0fg kmn 156 x'G5 eg] tL k"0ff{{ª\sx¿ s'g
s'g x'g\ <
The product of two consecutive positive integers is 156. Find the two numbers.
(d) nuftf/ cfpg] b'O{ cf]6f wgfTds lahf]/ ;ª\Vofx¿sf] u'0fg kmn 255 5 eg] tL
;ª\Vofx¿ kTtf
nufpg'xf];\ .
(The product of two consecutive positive odd numbers is 255. Find the two odd numbers.)
(e) Pskl5 csf{] cfpg] b'O{ cf]6f wgfTds hf]/ ;ª\Vofx¿sf] u'0fgkmn 288 5 eg] tL ;ª\Vofx¿
kTtf nufpg'xf];\ .
(f)
The product of two consecutive positive even numbers is 288. Find the number.
b'O{ cf]6f wgfTds ;ª\Vofsf] cGt/ 3 5 . olb tL ;ª\Vofsf ju{x¿sf] of]u
;ª\Vofx¿ kTtf nufpg'xf];\ .
kmn
89
eP tL
The difference between two positive numbers is 3. Find the numbers if the sum of their sttuares is
89.
pTt/x¿
gd'gf
(a) 7, 9
(d) 15, 17
(b) 12, 14
(e) 16, 18
3:
(c)12, 13
(f) 5, 8
b'O{ cª\ssf] ;ª\Vofdf cª\sx¿sf] u'0fg kmn 18 5 . pSt ;ª\Vofsf] cª\sx¿sf] :yfgdfg kl/jt{g
ubf{ aGg] ;ª\Vof ;'?sf] ;ª\VofeGbf 27 n] a9L x'G5 eg] ;'?sf] ;ª\Vof kTtf nufpg'xf];\ .
;dfwfg
b'O{ cª\sx¿ j|mdzM
X
/
Y dfgf}+
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
168
b'O{ cª\sx¿n] ag]sf ;ª\Vof= 10x+y
:yfgdf kl/jt{g ubf{ aGg] ;ª\Vof= 10y+X
ta, k|Zgsf] klxnf] ;t{cg';f/,
x × y=18
or y =
………………(1)
ta, k|Zgsf] bf];|f] ;t{cg';f/
x =(10 x +y)+27
or, 10y+ x -10-y=27
or, 9y-9 x =27
or, 9(y- x)=27
or, y- x =3
or, 10y+
or,
or,
−
=3
=3
or, 18 - x = 3x
2
or,0 = x 2+3x -18
x 2+6x-2x -18=0
or, x (x +6)-3(x + 6)=0
or, (x +6) (x -3) =0
Either, x +6 =0 or
or,
x = -6( x'“b}g)
x -3=0
x sf] dfg ;=s= (1) df /fVbf
y=
=6
x = 3 / y=6
pSt ;ª\Vof
3. cEof;sf
(a)
(b)
(c)
(d)
x =3
=10x-y =10 × 3 +6 =36
nflu k|Zgx¿
b'O{ cª\ssf] Pp6f ;ª\Vofdf cª\sx¿sf] u'0fg kmn 18 5 . pSt ;ª\Vofaf6 63 36fp“bf ;f]
;ª\Vofsf cª\sx¿ ablnG5g\ eg] ;ª\Vof kTtf nufpg'xf];\ .
b'O{ cª\sx¿sf] ;ª\Vof df cª\sx¿sf] u'0fg kmn 24 5 . olb Tof] ;ª\Vofdf 45 hf]l8of] eg]
;ª\Vof¿sf] cª\sx¿sf] :yfg kl/jt{g x'G5 eg] Tof] ;ª\Vof kTTff nufpg'xf];\ .
b'O{ cª\sx¿sf] Pp6f ;ª\Vofdf cª\sx¿sf] u'0fg kmn 8 5 . To; ;ª\Vofdf 18 yKbf ;f]
;ª\Vofsf cª\sxsf] :yfg ablnG5 eg] pSt ;ª\Vof kTtf nufpg'xf];\ .
b'O{ cª\sx¿ ldnL ag]sf] Pp6f ;ª\Vof 5 . tL cª\sx¿sf] of]u kmn 16 x'G5 To; ;ª\Vofdf 18
36fp“bf cª\sx¿sf] :yfg ablnG5 eg] ;f] ;ª\Vof kTtf nufpg'xf];\ .
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
169
(e)
Pp6f ;ª\Vof b'O{ cª\sn] ag]sf] 5 h;sf] of]u kmn 9 x'G5 olb ;f] ;ª\Vofsf] 3 u'0ff pSt
;ª\Vofsf] :yfg abNbf aGg] ;ª\Vofsf] 8 u'0ff;“u a/fa/ x'G5 eg] ;f] ;ª\Vof kTtf nufpg'xf];\ .
pTt/x¿
(a) 92
gd'gf
(a)
(b) 38
4:
(c) 24
xfn afa' / 5f]/fsf] pd]/ j|mdzM
210 lyof] kTtf nufpg'xf];\ .
(d) 97
35
(e) 72
/ 12 jif{ 5 . slt jif{cl3 ltgLx¿sf] pd]/sf] u'0fg kmn
;dfwfg M
pTt/, dfgf“}= x jif{cl3 afa' / 5f]/fsf] pd]/sf] u'0fg kmn
ta, k|Zgsf] cg';f/,
210
lyof] elg dfgf}+ .
(35- x) (12- x) =210
or, 420-35x -12x +
x 2-210=0
x 2-47x +210=0
or, x 2 -(42+5)x +210=0
or, x 2 -42x -5x +210=0
or, x (x -42)-5(x -42)=0
or, (x -42) (x -5)=0
Either, x -42=0
or, x - 5=0
x =42
x =5
x=42 jif{ x'“b}g lsgsL 42 jif{cl3 afa' g} hGd]sf lyPgg\ .
or,
∴ 5 jif{cl3
4.
(a)
(b)
(c)
(d)
(e)
plgx¿sf] pd]/sf] u'0fg kmn
cEof;sf nflu k|Zgx¿
210 x'G5
.
b'O{ lbbL alxgLsf] xfnsf] pd]/sf] u'0fg kmn 150 5 . 5 jif{cl3 lbbLsf] pd]/ alxgLsf] pd]/sf]
bf]Aa/ eP ltgLx¿sf] xfnsf] pd]/ kTtf nufpg'xf];\ .
bfh' / efOsf] pd]/sf] cGt/ 4 jif{ 5 / ltgLx¿sf] pd]/sf] u'0fg kmn 221 x'G5 eg] tL b'O{
efOsf] pd/] kTtf nfupg'xf];\ .
;Ltfsf] 4 jif{ cufl8 / 8 jif{kl5sf pd]/x¿sf] u'0fg kmn 28 eP pgsf] clxn]sf] pd]/ kTtf
nufpg'xf];\ .
b'O{ bfh' efOsf] xfnsf] pd]/sf] u'0fg kmn 160 5 . 4 jif{cl3 bfh'sf] pd]/ efOsf] pd]/sf] bf]Aa/
eP ltgLx¿sf] xfnsf] pd]/ kTtf nufpg'xf];\ .
Ps jif{cl3 Pp6f dflg; cfkm\gf] 5f]/feGbf 8 u'0ffn] h]7f] lyof] . clxn] p;sf] pd]/ 5f]/fsf]
pd]/sf] ju{;“u a/fa/ 5 . ltgLx¿sf] xfnsf] pd]/ kTtf nufpg'xf];\ .
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
170
(f)
afa' / 5f]/fsf] clxn]sf] pd]/ j|mdzM
lyof] xf]nf <
pTt/x¿
(a) 15 yrs., 10 yrs.
(d) 16 yrs, 10 yrs.
gd'gf
37
/ 8 slt jif{ klxn] ltgLx¿sf] pd]/sf] u'0fg kmn
(b) 16 yrs, 10 yrs.
(e) 7 yrs, 49 yrs.
5:
Pp6f cfotfsf/ hUufsf] If]qkmn
lgsfNg'xf];\ .
;dfwfg M
660 ju{
(c) 6 yrs,
(f) 5 yrs.
ld6/ / o;sf] kl/ldlt
104 eP
hUufsf] nDafO / rf}8fO
cfotsf/ hUufsf] nDafO l / rf}8fO b dfgf}“ .
ta, k|Zgsf] klxnf] ;t{cg';f/,
l × b =660
or, b =
(1)
ta, k|Zgsf] bf];|f] ;t{cg';f/,
2( + ) = 104
, + = 52
660
, +
= 52
,
+ 660
,
,
,
,
,
,
,
,
= 52
+ 660 = 52
− 52 + 660 = 0
− 30 − 22 + 660 = 0
+ 660 = 52
− 52 + 660 = 0
− 30 − 22 + 660 = 0
( − 30) − 22( − 30) = 0
( − 30)( − 22) = 0
( − 30) = 0
ℎ
or, ( − 22) = 0
∴ = 30
∴ = 22
2 sf] dfg ;=s= (1) df /fVbf
= 30
/fVbf
∴ = 30
− 22
or∴ = 22
/fVbf, = 30 = = 30
∴ = 22
/ = 30
x'g ;Sb}g lsgls nDafOeGbf rf}8fO nfdf] x'g ;Sb}g .
∴ pSt cfotsf/ hUufsf] nDafO / rf}8fO j|mdzM 30 /22 /x]5 .
k'gM
= 22
=
96
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
171
5. cEof;sf
(a)
nflu k|Zgx¿
olb Pp6f cfotsf/ rf}/sf] kl/ldlt 36 ld6/ / If]qkmn 77 ju{ ld6/ eP ;f] rf}/sf] nDafO /
rf}8fO kTtf nufpg'xf];\ .
(b) Pp6f sf]7fsf] nDafO To; sf]7fsf] rf}8fOeGbf 2 ld= nfdf] 5 . olb sf]7fsf] If]qkmn 63 ju{ ld=
5 eg] sf]7fsf] nDafO / rf}8fO kTtf nufpg'xf];\ .
(c) Pp6f cfotfsf/ hUufsf] rf}8fO nDafOeGbf 3 ld=eGbf sd 5 . olb pSt hUufsf] If]qkmn 88
ju{ ld= eP kl/ldlt kTtf nufpg'xf];\ .
(d) Pp6f sf]7fsf] If]qkmn 45 ju{ ld= 5 . olb nDafOdf 3 ld6/ sd / rf}8fOdf 1 ld6/ a9fp“bf
ju{ aG5 eg] pSt sf]7fsf] nDafO / rf}8fO kTtf nufpg'xf];\ .
(e) Pp6f sf]7fsf] If]qkmn 28 ju{ ld6/ 5 . olb nDafOdf 1 ld6/ sd / rf}8fOdf 2 ld6/ a9fp“bf
ju{ aG5 eg] pSt sf]7fsf] nDafO / rf}8fO kTtf nufpg'xf];\ .
(f) olb Pp6f cfotsf/ hUufsf] kl/ldlt 104 ld6/ / If]qkmn 640 ju{ ld6/ 5 eg] o;sf] nDafO
/ rf}8fO kTtf nufpg'xf];\ .
pTt/x¿
(a) 11m, 7m
(d) 9m, 5m
gd'gf
(b) 9m, 7 m (c) 38 m
(e) 7m, 4m
(f) 32m, 20m
6:
Pp6f ;dsf]0f lqe'hsf] s0f{ e'hf 29 ;]=ld 5 / c¿ b'O{ e'hfx¿sf] km/s 1 ;]=ld= 5 eg] tL e'hfx¿
kTtf nufpg'xf];\ .
;dfwfg M
oxfF, ;dsf]0f lqe'hsf] s0f{ h / cGo b'O{ e'hfx¿ j|mdzM p / b 5g\ egL dfgf}F .
ta, h = 29 cm. / P-b = 1 cm.
or, p = (l+b) cm. ______________(1)
xfdLnfO{ yfxf 5,
+b = ℎ [kfOyfuf]/; ;fWocg';f/]
or, (l+b)2+b2=(29)2-840=0
or, 2b2 +2b - 840=0
or, 2(b2+b-420)=0
or, b2+b-420=0
or, b2 +21b-20b-420=0
or, b (b+21)-20(b+21)=0
or, (b+21) (b-20) =0
Either,
or,
b+21=0
b-20=0
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
172
b = - 21cm x“'b}g
b=20cm
(1) df /fVbf
p = 1+20 = 21cm
pSt b'O{ e'hfx¿ 21cm / 20cm /x]5g\
dfg ;=s=
6. cEof;sf
(a)
nflu k|Zgx?
.
Pp6f ;dsf]0f lqe'hsf] s0f{ e'hf 5f]6f] e'hfsf] bf]Aa/eGbf klg 6 ld= nfdf] 5 . olb t];|f] e'hf
s0f{eGbf 2 ld6/ 5f]6f] 5 eg] lqe'hsf e'hfx¿ kTtf nufpg'xf];\ .
(b)
Pp6f ;dsf]0fL lqe'hsf] s0f{ To;sf] cfwf/eGbf 2 cm a9L / prfOsf] bf]Aa/eGbf 1 cm a9L
5 eg] lqe'hsf k|To]s e'hf kTtf nufpg'xf];\ .
(c)
Pp6f ;dsf]0fL lqe'hsf ;dsf]0f agfpg] e'hfx¿ To;sf] s0f{eGbf 5 cm /10 cm j|mdzM sd
5g\ eg] lqe'hsf e'hfx¿sf] nDafO kTtf nufpg'xf];\ .
pTt/x¿
(a) 10 cm, 24 cm, 26 cm
(b) 8cm, 15cm, 17cm
(c) 25cm, 20cm, 15cm
z}lIfs hgzlSt ljsf; s]Gb|, ;fgf]l7dL, eStk'/åf/f k|sflzt
ædfWolds txsf v'nf ljb\ofnosf l;sf?sf nflu cEof; k'l:tsfÆ
173