NCERT पाठ्यपुस्तक पाठ 4 - द्विघात समीकरण, कक्षा 10, गणित Class 10 Notes | EduRev

गणित कक्षा 10

Class 10 : NCERT पाठ्यपुस्तक पाठ 4 - द्विघात समीकरण, कक्षा 10, गणित Class 10 Notes | EduRev

 Page 1


78 xf.kr
4
4.1 Hkwfedk
vè;k; 2 esa] vkius fofHkUu izdkj osQ cgqinksa dk vè;;u fd;k gSA ax
2
 + bx + c, a ? 0 ,d
izdkj dk f}?kkr cgqin FkkA tc ge bl cgqin dks 'kwU; osQ rqY; dj nsrs gSa] rks gesa ,d
f}?kkr lehdj.k izkIr gks tkrh gSA okLrfod thou ls lacaf/r dbZ leL;kvksa dks gy
djus esa ge f}?kkr lehdj.kksa dk iz;ksx djrs gSaA mnkgj.kkFkZ] eku yhft, fd ,d /ekZFkZ
VªLV 300 oxZ ehVj {ks=kiQy dk izkFkZuk d{k cukuk
pkgrk gS] ftldh yackbZ mldh pkSM+kbZ osQ nks xqus ls
,d ehVj vf/d gksA d{k dh yackbZ vkSj pkSM+kbZ
D;k gksuh pkfg,\ ekuk d{k dh pkSM+kbZ
x ehVj gSA rc] mldh yackbZ ( 2x + 1) ehVj gksuh
pkfg,A ge bl lwpuk dks fp=kh; :i esa
vko`Qfr 4-1 tSlk fn[kk ldrs gSaA
vc d{k dk {ks=kiQy = (2x + 1). x m
2
 = (2x
2
 + x) m
2
blfy, 2x
2
 + x = 300 (fn;k gS)
vr% 2x
2
 + x – 300 = 0
blfy,] d{k dh pkSM+kbZ] lehdj.k 2x
2
 + x – 300 = 0, tks ,d f}?kkr lehdj.k gS] dks
larq"V djuk pkfg,A
vf/dka'k yksx fo'okl djrs gSa fd cschyksuokfl;ksa us gh loZizFke f}?kkr
lehdj.kksa dks gy fd;k FkkA mnkgj.k osQ fy,] os tkurs Fks fd oSQls nks la[;kvksa dks Kkr
fd;k tk ldrk gS] ftudk ;ksx rFkk xq.kuiQy fn;k gksA è;ku nhft, fd ;g leL;k
f}?kkr lehdj.k
vko`Qfr 4.1
2018-19
Page 2


78 xf.kr
4
4.1 Hkwfedk
vè;k; 2 esa] vkius fofHkUu izdkj osQ cgqinksa dk vè;;u fd;k gSA ax
2
 + bx + c, a ? 0 ,d
izdkj dk f}?kkr cgqin FkkA tc ge bl cgqin dks 'kwU; osQ rqY; dj nsrs gSa] rks gesa ,d
f}?kkr lehdj.k izkIr gks tkrh gSA okLrfod thou ls lacaf/r dbZ leL;kvksa dks gy
djus esa ge f}?kkr lehdj.kksa dk iz;ksx djrs gSaA mnkgj.kkFkZ] eku yhft, fd ,d /ekZFkZ
VªLV 300 oxZ ehVj {ks=kiQy dk izkFkZuk d{k cukuk
pkgrk gS] ftldh yackbZ mldh pkSM+kbZ osQ nks xqus ls
,d ehVj vf/d gksA d{k dh yackbZ vkSj pkSM+kbZ
D;k gksuh pkfg,\ ekuk d{k dh pkSM+kbZ
x ehVj gSA rc] mldh yackbZ ( 2x + 1) ehVj gksuh
pkfg,A ge bl lwpuk dks fp=kh; :i esa
vko`Qfr 4-1 tSlk fn[kk ldrs gSaA
vc d{k dk {ks=kiQy = (2x + 1). x m
2
 = (2x
2
 + x) m
2
blfy, 2x
2
 + x = 300 (fn;k gS)
vr% 2x
2
 + x – 300 = 0
blfy,] d{k dh pkSM+kbZ] lehdj.k 2x
2
 + x – 300 = 0, tks ,d f}?kkr lehdj.k gS] dks
larq"V djuk pkfg,A
vf/dka'k yksx fo'okl djrs gSa fd cschyksuokfl;ksa us gh loZizFke f}?kkr
lehdj.kksa dks gy fd;k FkkA mnkgj.k osQ fy,] os tkurs Fks fd oSQls nks la[;kvksa dks Kkr
fd;k tk ldrk gS] ftudk ;ksx rFkk xq.kuiQy fn;k gksA è;ku nhft, fd ;g leL;k
f}?kkr lehdj.k
vko`Qfr 4.1
2018-19
f}?kkr lehdj.k 79
x
2
 – px + q = 0 osQ izdkj osQ lehdj.k dks gy djus osQ rqY; gSA ;wukuh xf.krK ;wfDyM
us yackb;k¡ Kkr djus dh ,d T;kferh; fof/ fodflr dh ftldks ge orZeku 'kCnkoyh
eas f}?kkr lehdj.k osQ gy dgrs gSaA O;kid :i esa] f}?kkr lehdj.kksa dks gy djus dk
Js; cgq/k izkphu Hkkjrh; xf.krKksa dks tkrk gSA okLro esa] czãxqIr (lk-;q- 598-665) us
ax
2
 + bx = c osQ :i osQ f}?kkr lehdj.k dks gy djus dk ,d Li"V lw=k fn;k FkkA ckn
esa] Jh/jkpk;Z (lk-;q- 1025) us ,d lw=k izfrikfnr fd;k] ftls vc f}?kkrh lw=k osQ :i esa
tkuk tkrk gS] tks iw.kZ oxZ fof/ ls f}?kkr lehdj.k dks gy djus ij izkIr gqvk (tSlk
HkkLdj II us fy[kk)A ,d vjc xf.krK vy&[okfjT+eh (yxHkx lk-;q- 800) us Hkh fofHkUu
izdkj osQ f}?kkr lehdj.kksa dk vè;;u fd;kA vczkg~e ckj fgÕ;k gk&uklh ;wjks us 1145
esa Nih viuh iqLrd ^fycj backMksje* esa fofHkUu f}?kkr lehdj.kksa osQ iw.kZ gy fn,A
bl vè;k; esa] vki f}?kkr lehdj.kksa vkSj muosQ gy Kkr djus dh fofHkUu fofèk;ksa
dk vè;;u djsaxsA nSfud thou dh dbZ fLFkfr;ksa esa Hkh vki f}?kkr lehdj.kksa osQ oqQN
mi;ksx ns[ksaxsA
4.2 f}?kkr lehdj.k
pj x esa ,d f}?kkr lehdj.k a x
2
 + b x + c = 0 osQ izdkj dh gksrh gS] tgk¡ a, b, c okLrfod
la[;k,¡ gSa rFkk a ? 0 gSA mnkgj.k osQ fy,] 2x
2
 + x – 300 = 0 ,d f}?kkr lehdj.k gSA blh
izdkj]  2x
2
 – 3x + 1 = 0, 4x – 3x
2
 + 2 = 0 vkSj 1 – x
2
 + 300 = 0 Hkh f}?kkr lehdj.k gSaA
okLro esa] dksbZ Hkh lehdj.k p(x) = 0, tgk¡ p(x), ?kkr 2 dk ,d cgqin gS] ,d
f}?kkr lehdj.k dgykrh gSA ijarq tc ge p(x) osQ in ?kkrksa osQ ?kVrs Øe esa fy[krs gSa]
rks gesa lehdj.k dk ekud :i izkIr gksrk gSA vFkkZr~  ax
2
 + bx + c = 0, a ? 0,
f}?kkr lehdj.k dk ekud :i dgykrk gSA
f}?kkr lehdj.k gekjs vklikl osQ ifjos'k dh vusd fLFkfr;ksa ,oa xf.kr osQ fofHkUu
{ks=kksa esa iz;qDr gksrs gSaA vkb, ge oqQN mnkgj.k ysaA
mnkgj.k 1 : fuEu fLFkfr;ksa dks xf.krh; :i esa O;Dr dhft, %
(i) tkWu vkSj thoarh nksuksa osQ ikl oqQy feykdj 45 oaQps gSaA nksuksa ik¡p&ik¡p oaQps [kks nsrs gSa
vkSj vc muosQ ikl oaQpksa dh la[;k dk xq.kuiQy 124 gSA ge tkuuk pkgsaxs fd vkjaHk
esa muosQ ikl fdrus&fdrus oaQps FksA
(ii) ,d oqQVhj m|ksx ,d fnu esa oqQN f[kykSus fu£er djrk gSA izR;sd f[kykSus dk ewY;
(` esa) 55 esa ls ,d fnu esa fuekZ.k fd, x, f[kykSus dh la[;k dks ?kVkus ls
2018-19
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78 xf.kr
4
4.1 Hkwfedk
vè;k; 2 esa] vkius fofHkUu izdkj osQ cgqinksa dk vè;;u fd;k gSA ax
2
 + bx + c, a ? 0 ,d
izdkj dk f}?kkr cgqin FkkA tc ge bl cgqin dks 'kwU; osQ rqY; dj nsrs gSa] rks gesa ,d
f}?kkr lehdj.k izkIr gks tkrh gSA okLrfod thou ls lacaf/r dbZ leL;kvksa dks gy
djus esa ge f}?kkr lehdj.kksa dk iz;ksx djrs gSaA mnkgj.kkFkZ] eku yhft, fd ,d /ekZFkZ
VªLV 300 oxZ ehVj {ks=kiQy dk izkFkZuk d{k cukuk
pkgrk gS] ftldh yackbZ mldh pkSM+kbZ osQ nks xqus ls
,d ehVj vf/d gksA d{k dh yackbZ vkSj pkSM+kbZ
D;k gksuh pkfg,\ ekuk d{k dh pkSM+kbZ
x ehVj gSA rc] mldh yackbZ ( 2x + 1) ehVj gksuh
pkfg,A ge bl lwpuk dks fp=kh; :i esa
vko`Qfr 4-1 tSlk fn[kk ldrs gSaA
vc d{k dk {ks=kiQy = (2x + 1). x m
2
 = (2x
2
 + x) m
2
blfy, 2x
2
 + x = 300 (fn;k gS)
vr% 2x
2
 + x – 300 = 0
blfy,] d{k dh pkSM+kbZ] lehdj.k 2x
2
 + x – 300 = 0, tks ,d f}?kkr lehdj.k gS] dks
larq"V djuk pkfg,A
vf/dka'k yksx fo'okl djrs gSa fd cschyksuokfl;ksa us gh loZizFke f}?kkr
lehdj.kksa dks gy fd;k FkkA mnkgj.k osQ fy,] os tkurs Fks fd oSQls nks la[;kvksa dks Kkr
fd;k tk ldrk gS] ftudk ;ksx rFkk xq.kuiQy fn;k gksA è;ku nhft, fd ;g leL;k
f}?kkr lehdj.k
vko`Qfr 4.1
2018-19
f}?kkr lehdj.k 79
x
2
 – px + q = 0 osQ izdkj osQ lehdj.k dks gy djus osQ rqY; gSA ;wukuh xf.krK ;wfDyM
us yackb;k¡ Kkr djus dh ,d T;kferh; fof/ fodflr dh ftldks ge orZeku 'kCnkoyh
eas f}?kkr lehdj.k osQ gy dgrs gSaA O;kid :i esa] f}?kkr lehdj.kksa dks gy djus dk
Js; cgq/k izkphu Hkkjrh; xf.krKksa dks tkrk gSA okLro esa] czãxqIr (lk-;q- 598-665) us
ax
2
 + bx = c osQ :i osQ f}?kkr lehdj.k dks gy djus dk ,d Li"V lw=k fn;k FkkA ckn
esa] Jh/jkpk;Z (lk-;q- 1025) us ,d lw=k izfrikfnr fd;k] ftls vc f}?kkrh lw=k osQ :i esa
tkuk tkrk gS] tks iw.kZ oxZ fof/ ls f}?kkr lehdj.k dks gy djus ij izkIr gqvk (tSlk
HkkLdj II us fy[kk)A ,d vjc xf.krK vy&[okfjT+eh (yxHkx lk-;q- 800) us Hkh fofHkUu
izdkj osQ f}?kkr lehdj.kksa dk vè;;u fd;kA vczkg~e ckj fgÕ;k gk&uklh ;wjks us 1145
esa Nih viuh iqLrd ^fycj backMksje* esa fofHkUu f}?kkr lehdj.kksa osQ iw.kZ gy fn,A
bl vè;k; esa] vki f}?kkr lehdj.kksa vkSj muosQ gy Kkr djus dh fofHkUu fofèk;ksa
dk vè;;u djsaxsA nSfud thou dh dbZ fLFkfr;ksa esa Hkh vki f}?kkr lehdj.kksa osQ oqQN
mi;ksx ns[ksaxsA
4.2 f}?kkr lehdj.k
pj x esa ,d f}?kkr lehdj.k a x
2
 + b x + c = 0 osQ izdkj dh gksrh gS] tgk¡ a, b, c okLrfod
la[;k,¡ gSa rFkk a ? 0 gSA mnkgj.k osQ fy,] 2x
2
 + x – 300 = 0 ,d f}?kkr lehdj.k gSA blh
izdkj]  2x
2
 – 3x + 1 = 0, 4x – 3x
2
 + 2 = 0 vkSj 1 – x
2
 + 300 = 0 Hkh f}?kkr lehdj.k gSaA
okLro esa] dksbZ Hkh lehdj.k p(x) = 0, tgk¡ p(x), ?kkr 2 dk ,d cgqin gS] ,d
f}?kkr lehdj.k dgykrh gSA ijarq tc ge p(x) osQ in ?kkrksa osQ ?kVrs Øe esa fy[krs gSa]
rks gesa lehdj.k dk ekud :i izkIr gksrk gSA vFkkZr~  ax
2
 + bx + c = 0, a ? 0,
f}?kkr lehdj.k dk ekud :i dgykrk gSA
f}?kkr lehdj.k gekjs vklikl osQ ifjos'k dh vusd fLFkfr;ksa ,oa xf.kr osQ fofHkUu
{ks=kksa esa iz;qDr gksrs gSaA vkb, ge oqQN mnkgj.k ysaA
mnkgj.k 1 : fuEu fLFkfr;ksa dks xf.krh; :i esa O;Dr dhft, %
(i) tkWu vkSj thoarh nksuksa osQ ikl oqQy feykdj 45 oaQps gSaA nksuksa ik¡p&ik¡p oaQps [kks nsrs gSa
vkSj vc muosQ ikl oaQpksa dh la[;k dk xq.kuiQy 124 gSA ge tkuuk pkgsaxs fd vkjaHk
esa muosQ ikl fdrus&fdrus oaQps FksA
(ii) ,d oqQVhj m|ksx ,d fnu esa oqQN f[kykSus fu£er djrk gSA izR;sd f[kykSus dk ewY;
(` esa) 55 esa ls ,d fnu esa fuekZ.k fd, x, f[kykSus dh la[;k dks ?kVkus ls
2018-19
80 xf.kr
izkIr la[;k osQ cjkcj gSA fdlh ,d fnu] oqQy fuekZ.k ykxr ` 750 FkhA ge ml
fnu fuekZ.k fd, x, f[kykSuksa dh la[;k Kkr djuk pkgsaxsA
gy :
(i) ekuk fd tkWu osQ oaQpksa dh la[;k  x FkhA
rc thoarh osQ oaQpksa dh la[;k = 45 – x (D;ksa\)
tkWu osQ ikl] 5 oaQps [kks nsus osQ ckn] cps oaQpksa dh la[;k = x – 5
thoarh osQ ikl] 5 oaQps [kksus osQ ckn] cps oaQpksa dh la[;k = 45 – x – 5
= 40 – x
vr% mudk xq.kuiQy = (x – 5) (40 – x)
= 40x – x
2
 – 200 + 5x
= – x
2
 + 45x – 200
vc – x
2
 + 45x – 200 = 124 (fn;k gS fd xq.kuiQy = 124)
vFkkZr~ – x
2
 + 45x – 324 = 0
vFkkZr~ x
2
 – 45x + 324 = 0
vr% tkWu osQ ikl ftrus oaQps Fks] tks lehdj.k
x
2
 – 45x + 324 = 0
dks larq"V djrs gSaA
(ii) ekuk ml fnu fufeZr f[kykSuksa dh la[;k  x gSA
blfy,] ml fnu izR;sd f[kykSus dh fuekZ.k ykxr (#i;ksa esa) = 55 – x
vr%] ml fnu oqQy fuekZ.k ykxr (#i;ksa esa) = x (55 – x)
blfy, x (55 – x) = 750
vFkkZr~ 55x – x
2
 = 750
vFkkZr~ – x
2
 + 55x – 750 =0
vFkkZr~ x
2
 – 55x + 750 = 0
vr% ml fnu fuekZ.k fd, x, f[kykSuksa dh la[;k f}?kkr lehdj.k
x
2
 – 55x + 750 = 0
dks larq"V djrh gSA
2018-19
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78 xf.kr
4
4.1 Hkwfedk
vè;k; 2 esa] vkius fofHkUu izdkj osQ cgqinksa dk vè;;u fd;k gSA ax
2
 + bx + c, a ? 0 ,d
izdkj dk f}?kkr cgqin FkkA tc ge bl cgqin dks 'kwU; osQ rqY; dj nsrs gSa] rks gesa ,d
f}?kkr lehdj.k izkIr gks tkrh gSA okLrfod thou ls lacaf/r dbZ leL;kvksa dks gy
djus esa ge f}?kkr lehdj.kksa dk iz;ksx djrs gSaA mnkgj.kkFkZ] eku yhft, fd ,d /ekZFkZ
VªLV 300 oxZ ehVj {ks=kiQy dk izkFkZuk d{k cukuk
pkgrk gS] ftldh yackbZ mldh pkSM+kbZ osQ nks xqus ls
,d ehVj vf/d gksA d{k dh yackbZ vkSj pkSM+kbZ
D;k gksuh pkfg,\ ekuk d{k dh pkSM+kbZ
x ehVj gSA rc] mldh yackbZ ( 2x + 1) ehVj gksuh
pkfg,A ge bl lwpuk dks fp=kh; :i esa
vko`Qfr 4-1 tSlk fn[kk ldrs gSaA
vc d{k dk {ks=kiQy = (2x + 1). x m
2
 = (2x
2
 + x) m
2
blfy, 2x
2
 + x = 300 (fn;k gS)
vr% 2x
2
 + x – 300 = 0
blfy,] d{k dh pkSM+kbZ] lehdj.k 2x
2
 + x – 300 = 0, tks ,d f}?kkr lehdj.k gS] dks
larq"V djuk pkfg,A
vf/dka'k yksx fo'okl djrs gSa fd cschyksuokfl;ksa us gh loZizFke f}?kkr
lehdj.kksa dks gy fd;k FkkA mnkgj.k osQ fy,] os tkurs Fks fd oSQls nks la[;kvksa dks Kkr
fd;k tk ldrk gS] ftudk ;ksx rFkk xq.kuiQy fn;k gksA è;ku nhft, fd ;g leL;k
f}?kkr lehdj.k
vko`Qfr 4.1
2018-19
f}?kkr lehdj.k 79
x
2
 – px + q = 0 osQ izdkj osQ lehdj.k dks gy djus osQ rqY; gSA ;wukuh xf.krK ;wfDyM
us yackb;k¡ Kkr djus dh ,d T;kferh; fof/ fodflr dh ftldks ge orZeku 'kCnkoyh
eas f}?kkr lehdj.k osQ gy dgrs gSaA O;kid :i esa] f}?kkr lehdj.kksa dks gy djus dk
Js; cgq/k izkphu Hkkjrh; xf.krKksa dks tkrk gSA okLro esa] czãxqIr (lk-;q- 598-665) us
ax
2
 + bx = c osQ :i osQ f}?kkr lehdj.k dks gy djus dk ,d Li"V lw=k fn;k FkkA ckn
esa] Jh/jkpk;Z (lk-;q- 1025) us ,d lw=k izfrikfnr fd;k] ftls vc f}?kkrh lw=k osQ :i esa
tkuk tkrk gS] tks iw.kZ oxZ fof/ ls f}?kkr lehdj.k dks gy djus ij izkIr gqvk (tSlk
HkkLdj II us fy[kk)A ,d vjc xf.krK vy&[okfjT+eh (yxHkx lk-;q- 800) us Hkh fofHkUu
izdkj osQ f}?kkr lehdj.kksa dk vè;;u fd;kA vczkg~e ckj fgÕ;k gk&uklh ;wjks us 1145
esa Nih viuh iqLrd ^fycj backMksje* esa fofHkUu f}?kkr lehdj.kksa osQ iw.kZ gy fn,A
bl vè;k; esa] vki f}?kkr lehdj.kksa vkSj muosQ gy Kkr djus dh fofHkUu fofèk;ksa
dk vè;;u djsaxsA nSfud thou dh dbZ fLFkfr;ksa esa Hkh vki f}?kkr lehdj.kksa osQ oqQN
mi;ksx ns[ksaxsA
4.2 f}?kkr lehdj.k
pj x esa ,d f}?kkr lehdj.k a x
2
 + b x + c = 0 osQ izdkj dh gksrh gS] tgk¡ a, b, c okLrfod
la[;k,¡ gSa rFkk a ? 0 gSA mnkgj.k osQ fy,] 2x
2
 + x – 300 = 0 ,d f}?kkr lehdj.k gSA blh
izdkj]  2x
2
 – 3x + 1 = 0, 4x – 3x
2
 + 2 = 0 vkSj 1 – x
2
 + 300 = 0 Hkh f}?kkr lehdj.k gSaA
okLro esa] dksbZ Hkh lehdj.k p(x) = 0, tgk¡ p(x), ?kkr 2 dk ,d cgqin gS] ,d
f}?kkr lehdj.k dgykrh gSA ijarq tc ge p(x) osQ in ?kkrksa osQ ?kVrs Øe esa fy[krs gSa]
rks gesa lehdj.k dk ekud :i izkIr gksrk gSA vFkkZr~  ax
2
 + bx + c = 0, a ? 0,
f}?kkr lehdj.k dk ekud :i dgykrk gSA
f}?kkr lehdj.k gekjs vklikl osQ ifjos'k dh vusd fLFkfr;ksa ,oa xf.kr osQ fofHkUu
{ks=kksa esa iz;qDr gksrs gSaA vkb, ge oqQN mnkgj.k ysaA
mnkgj.k 1 : fuEu fLFkfr;ksa dks xf.krh; :i esa O;Dr dhft, %
(i) tkWu vkSj thoarh nksuksa osQ ikl oqQy feykdj 45 oaQps gSaA nksuksa ik¡p&ik¡p oaQps [kks nsrs gSa
vkSj vc muosQ ikl oaQpksa dh la[;k dk xq.kuiQy 124 gSA ge tkuuk pkgsaxs fd vkjaHk
esa muosQ ikl fdrus&fdrus oaQps FksA
(ii) ,d oqQVhj m|ksx ,d fnu esa oqQN f[kykSus fu£er djrk gSA izR;sd f[kykSus dk ewY;
(` esa) 55 esa ls ,d fnu esa fuekZ.k fd, x, f[kykSus dh la[;k dks ?kVkus ls
2018-19
80 xf.kr
izkIr la[;k osQ cjkcj gSA fdlh ,d fnu] oqQy fuekZ.k ykxr ` 750 FkhA ge ml
fnu fuekZ.k fd, x, f[kykSuksa dh la[;k Kkr djuk pkgsaxsA
gy :
(i) ekuk fd tkWu osQ oaQpksa dh la[;k  x FkhA
rc thoarh osQ oaQpksa dh la[;k = 45 – x (D;ksa\)
tkWu osQ ikl] 5 oaQps [kks nsus osQ ckn] cps oaQpksa dh la[;k = x – 5
thoarh osQ ikl] 5 oaQps [kksus osQ ckn] cps oaQpksa dh la[;k = 45 – x – 5
= 40 – x
vr% mudk xq.kuiQy = (x – 5) (40 – x)
= 40x – x
2
 – 200 + 5x
= – x
2
 + 45x – 200
vc – x
2
 + 45x – 200 = 124 (fn;k gS fd xq.kuiQy = 124)
vFkkZr~ – x
2
 + 45x – 324 = 0
vFkkZr~ x
2
 – 45x + 324 = 0
vr% tkWu osQ ikl ftrus oaQps Fks] tks lehdj.k
x
2
 – 45x + 324 = 0
dks larq"V djrs gSaA
(ii) ekuk ml fnu fufeZr f[kykSuksa dh la[;k  x gSA
blfy,] ml fnu izR;sd f[kykSus dh fuekZ.k ykxr (#i;ksa esa) = 55 – x
vr%] ml fnu oqQy fuekZ.k ykxr (#i;ksa esa) = x (55 – x)
blfy, x (55 – x) = 750
vFkkZr~ 55x – x
2
 = 750
vFkkZr~ – x
2
 + 55x – 750 =0
vFkkZr~ x
2
 – 55x + 750 = 0
vr% ml fnu fuekZ.k fd, x, f[kykSuksa dh la[;k f}?kkr lehdj.k
x
2
 – 55x + 750 = 0
dks larq"V djrh gSA
2018-19
f}?kkr lehdj.k 81
mnkgj.k 2 : tk¡p dhft, fd fuEu f}?kkr lehdj.k gSa ;k ugha%
(i) (x – 2)
2
 + 1 = 2x – 3 (ii) x(x + 1) + 8 = (x + 2) (x – 2)
(iii) x (2x + 3) = x
2
 + 1 (iv) (x + 2)
3
 = x
3
 – 4
gy :
(i) ck;k¡ i{k = (x – 2)
2
 + 1 = x
2
 – 4x + 4 + 1 = x
2
 – 4x + 5
blfy, (x – 2)
2 
+
 
1 = 2x – 3  dks
x
2
 – 4x + 5 = 2x – 3 fy[kk tk ldrk gSA
vFkkZr~ x
2
 – 6x + 8 = 0
; g a x
2
 + b x + c = 0 osQ izdkj dk gSA
vr% fn;k x;k lehdj.k ,d f}?kkr lehdj.k gSA
(ii) pw¡fd x(x + 1) + 8 = x
2
 + x + 8 vkSj (x + 2)(x – 2) = x
2
 – 4 gS]
blfy, x
2
 + x + 8 = x
2
 – 4
vFkkZr~ x + 12 = 0
;g ax
2
 + bx + c = 0 osQ izdkj dk lehdj.k ugha gSA blfy,] fn;k gqvk
lehdj.k ,d f}?kkr lehdj.k ugha gSA
(iii) ;gk¡ ck;k¡ i{k = x (2x + 3) = 2x
2
 + 3x
vr% x (2x + 3) = x
2
 + 1 dks fy[kk tk ldrk gS%
2x
2
 + 3x = x
2
 + 1
blfy, x
2
 + 3x – 1 = 0 gesa izkIr gksrk gSA
; g a x
2
 + b x + c = 0 osQ izdkj dk lehdj.k gSA
vr%] fn;k x;k lehdj.k ,d f}?kkr lehdj.k gSA
(iv) ;gk¡ ck;k¡ i{k = (x + 2)
3
 = x
3
 + 6x
2
 + 12x + 8
vr% (x + 2)
3
 = x
3
 – 4 dks fy[kk tk ldrk gS%
x
3
 + 6x
2
 + 12x + 8 = x
3
 – 4
vFkkZr~ 6x
2
 + 12x + 12 = 0 ;k x
2
 + 2x + 2 = 0
; g a x
2
 + b x + c = 0 osQ izdkj dk lehdj.k gSA
vr% fn;k x;k lehdj.k ,d f}?kkr lehdj.k gSA
2018-19
Page 5


78 xf.kr
4
4.1 Hkwfedk
vè;k; 2 esa] vkius fofHkUu izdkj osQ cgqinksa dk vè;;u fd;k gSA ax
2
 + bx + c, a ? 0 ,d
izdkj dk f}?kkr cgqin FkkA tc ge bl cgqin dks 'kwU; osQ rqY; dj nsrs gSa] rks gesa ,d
f}?kkr lehdj.k izkIr gks tkrh gSA okLrfod thou ls lacaf/r dbZ leL;kvksa dks gy
djus esa ge f}?kkr lehdj.kksa dk iz;ksx djrs gSaA mnkgj.kkFkZ] eku yhft, fd ,d /ekZFkZ
VªLV 300 oxZ ehVj {ks=kiQy dk izkFkZuk d{k cukuk
pkgrk gS] ftldh yackbZ mldh pkSM+kbZ osQ nks xqus ls
,d ehVj vf/d gksA d{k dh yackbZ vkSj pkSM+kbZ
D;k gksuh pkfg,\ ekuk d{k dh pkSM+kbZ
x ehVj gSA rc] mldh yackbZ ( 2x + 1) ehVj gksuh
pkfg,A ge bl lwpuk dks fp=kh; :i esa
vko`Qfr 4-1 tSlk fn[kk ldrs gSaA
vc d{k dk {ks=kiQy = (2x + 1). x m
2
 = (2x
2
 + x) m
2
blfy, 2x
2
 + x = 300 (fn;k gS)
vr% 2x
2
 + x – 300 = 0
blfy,] d{k dh pkSM+kbZ] lehdj.k 2x
2
 + x – 300 = 0, tks ,d f}?kkr lehdj.k gS] dks
larq"V djuk pkfg,A
vf/dka'k yksx fo'okl djrs gSa fd cschyksuokfl;ksa us gh loZizFke f}?kkr
lehdj.kksa dks gy fd;k FkkA mnkgj.k osQ fy,] os tkurs Fks fd oSQls nks la[;kvksa dks Kkr
fd;k tk ldrk gS] ftudk ;ksx rFkk xq.kuiQy fn;k gksA è;ku nhft, fd ;g leL;k
f}?kkr lehdj.k
vko`Qfr 4.1
2018-19
f}?kkr lehdj.k 79
x
2
 – px + q = 0 osQ izdkj osQ lehdj.k dks gy djus osQ rqY; gSA ;wukuh xf.krK ;wfDyM
us yackb;k¡ Kkr djus dh ,d T;kferh; fof/ fodflr dh ftldks ge orZeku 'kCnkoyh
eas f}?kkr lehdj.k osQ gy dgrs gSaA O;kid :i esa] f}?kkr lehdj.kksa dks gy djus dk
Js; cgq/k izkphu Hkkjrh; xf.krKksa dks tkrk gSA okLro esa] czãxqIr (lk-;q- 598-665) us
ax
2
 + bx = c osQ :i osQ f}?kkr lehdj.k dks gy djus dk ,d Li"V lw=k fn;k FkkA ckn
esa] Jh/jkpk;Z (lk-;q- 1025) us ,d lw=k izfrikfnr fd;k] ftls vc f}?kkrh lw=k osQ :i esa
tkuk tkrk gS] tks iw.kZ oxZ fof/ ls f}?kkr lehdj.k dks gy djus ij izkIr gqvk (tSlk
HkkLdj II us fy[kk)A ,d vjc xf.krK vy&[okfjT+eh (yxHkx lk-;q- 800) us Hkh fofHkUu
izdkj osQ f}?kkr lehdj.kksa dk vè;;u fd;kA vczkg~e ckj fgÕ;k gk&uklh ;wjks us 1145
esa Nih viuh iqLrd ^fycj backMksje* esa fofHkUu f}?kkr lehdj.kksa osQ iw.kZ gy fn,A
bl vè;k; esa] vki f}?kkr lehdj.kksa vkSj muosQ gy Kkr djus dh fofHkUu fofèk;ksa
dk vè;;u djsaxsA nSfud thou dh dbZ fLFkfr;ksa esa Hkh vki f}?kkr lehdj.kksa osQ oqQN
mi;ksx ns[ksaxsA
4.2 f}?kkr lehdj.k
pj x esa ,d f}?kkr lehdj.k a x
2
 + b x + c = 0 osQ izdkj dh gksrh gS] tgk¡ a, b, c okLrfod
la[;k,¡ gSa rFkk a ? 0 gSA mnkgj.k osQ fy,] 2x
2
 + x – 300 = 0 ,d f}?kkr lehdj.k gSA blh
izdkj]  2x
2
 – 3x + 1 = 0, 4x – 3x
2
 + 2 = 0 vkSj 1 – x
2
 + 300 = 0 Hkh f}?kkr lehdj.k gSaA
okLro esa] dksbZ Hkh lehdj.k p(x) = 0, tgk¡ p(x), ?kkr 2 dk ,d cgqin gS] ,d
f}?kkr lehdj.k dgykrh gSA ijarq tc ge p(x) osQ in ?kkrksa osQ ?kVrs Øe esa fy[krs gSa]
rks gesa lehdj.k dk ekud :i izkIr gksrk gSA vFkkZr~  ax
2
 + bx + c = 0, a ? 0,
f}?kkr lehdj.k dk ekud :i dgykrk gSA
f}?kkr lehdj.k gekjs vklikl osQ ifjos'k dh vusd fLFkfr;ksa ,oa xf.kr osQ fofHkUu
{ks=kksa esa iz;qDr gksrs gSaA vkb, ge oqQN mnkgj.k ysaA
mnkgj.k 1 : fuEu fLFkfr;ksa dks xf.krh; :i esa O;Dr dhft, %
(i) tkWu vkSj thoarh nksuksa osQ ikl oqQy feykdj 45 oaQps gSaA nksuksa ik¡p&ik¡p oaQps [kks nsrs gSa
vkSj vc muosQ ikl oaQpksa dh la[;k dk xq.kuiQy 124 gSA ge tkuuk pkgsaxs fd vkjaHk
esa muosQ ikl fdrus&fdrus oaQps FksA
(ii) ,d oqQVhj m|ksx ,d fnu esa oqQN f[kykSus fu£er djrk gSA izR;sd f[kykSus dk ewY;
(` esa) 55 esa ls ,d fnu esa fuekZ.k fd, x, f[kykSus dh la[;k dks ?kVkus ls
2018-19
80 xf.kr
izkIr la[;k osQ cjkcj gSA fdlh ,d fnu] oqQy fuekZ.k ykxr ` 750 FkhA ge ml
fnu fuekZ.k fd, x, f[kykSuksa dh la[;k Kkr djuk pkgsaxsA
gy :
(i) ekuk fd tkWu osQ oaQpksa dh la[;k  x FkhA
rc thoarh osQ oaQpksa dh la[;k = 45 – x (D;ksa\)
tkWu osQ ikl] 5 oaQps [kks nsus osQ ckn] cps oaQpksa dh la[;k = x – 5
thoarh osQ ikl] 5 oaQps [kksus osQ ckn] cps oaQpksa dh la[;k = 45 – x – 5
= 40 – x
vr% mudk xq.kuiQy = (x – 5) (40 – x)
= 40x – x
2
 – 200 + 5x
= – x
2
 + 45x – 200
vc – x
2
 + 45x – 200 = 124 (fn;k gS fd xq.kuiQy = 124)
vFkkZr~ – x
2
 + 45x – 324 = 0
vFkkZr~ x
2
 – 45x + 324 = 0
vr% tkWu osQ ikl ftrus oaQps Fks] tks lehdj.k
x
2
 – 45x + 324 = 0
dks larq"V djrs gSaA
(ii) ekuk ml fnu fufeZr f[kykSuksa dh la[;k  x gSA
blfy,] ml fnu izR;sd f[kykSus dh fuekZ.k ykxr (#i;ksa esa) = 55 – x
vr%] ml fnu oqQy fuekZ.k ykxr (#i;ksa esa) = x (55 – x)
blfy, x (55 – x) = 750
vFkkZr~ 55x – x
2
 = 750
vFkkZr~ – x
2
 + 55x – 750 =0
vFkkZr~ x
2
 – 55x + 750 = 0
vr% ml fnu fuekZ.k fd, x, f[kykSuksa dh la[;k f}?kkr lehdj.k
x
2
 – 55x + 750 = 0
dks larq"V djrh gSA
2018-19
f}?kkr lehdj.k 81
mnkgj.k 2 : tk¡p dhft, fd fuEu f}?kkr lehdj.k gSa ;k ugha%
(i) (x – 2)
2
 + 1 = 2x – 3 (ii) x(x + 1) + 8 = (x + 2) (x – 2)
(iii) x (2x + 3) = x
2
 + 1 (iv) (x + 2)
3
 = x
3
 – 4
gy :
(i) ck;k¡ i{k = (x – 2)
2
 + 1 = x
2
 – 4x + 4 + 1 = x
2
 – 4x + 5
blfy, (x – 2)
2 
+
 
1 = 2x – 3  dks
x
2
 – 4x + 5 = 2x – 3 fy[kk tk ldrk gSA
vFkkZr~ x
2
 – 6x + 8 = 0
; g a x
2
 + b x + c = 0 osQ izdkj dk gSA
vr% fn;k x;k lehdj.k ,d f}?kkr lehdj.k gSA
(ii) pw¡fd x(x + 1) + 8 = x
2
 + x + 8 vkSj (x + 2)(x – 2) = x
2
 – 4 gS]
blfy, x
2
 + x + 8 = x
2
 – 4
vFkkZr~ x + 12 = 0
;g ax
2
 + bx + c = 0 osQ izdkj dk lehdj.k ugha gSA blfy,] fn;k gqvk
lehdj.k ,d f}?kkr lehdj.k ugha gSA
(iii) ;gk¡ ck;k¡ i{k = x (2x + 3) = 2x
2
 + 3x
vr% x (2x + 3) = x
2
 + 1 dks fy[kk tk ldrk gS%
2x
2
 + 3x = x
2
 + 1
blfy, x
2
 + 3x – 1 = 0 gesa izkIr gksrk gSA
; g a x
2
 + b x + c = 0 osQ izdkj dk lehdj.k gSA
vr%] fn;k x;k lehdj.k ,d f}?kkr lehdj.k gSA
(iv) ;gk¡ ck;k¡ i{k = (x + 2)
3
 = x
3
 + 6x
2
 + 12x + 8
vr% (x + 2)
3
 = x
3
 – 4 dks fy[kk tk ldrk gS%
x
3
 + 6x
2
 + 12x + 8 = x
3
 – 4
vFkkZr~ 6x
2
 + 12x + 12 = 0 ;k x
2
 + 2x + 2 = 0
; g a x
2
 + b x + c = 0 osQ izdkj dk lehdj.k gSA
vr% fn;k x;k lehdj.k ,d f}?kkr lehdj.k gSA
2018-19
82 xf.kr
fVIi.kh : è;ku nhft, fd mi;qZDr (ii) esa] fn;k x;k lehdj.k ns[kus esa f}?kkr lehdj.k
yxrk gS] ijarq ;g f}?kkr lehdj.k ugha gSA
mi;qZDr (iv) esa] lehdj.k ns[kus esa f=k?kkr (?kkr 3 dk lehdj.k) yxrk gS vkSj f}?kkr
ugha yxrk gSA ijarq og f}?kkr lehdj.k fudyrk gSA tSlk vki ns[krs gSa lehdj.k dks ;g
r; djus fd og f}?kkr gS vFkok ugha] geas mldk ljyhdj.k djuk vko';d gSA
iz'ukoyh 4.1
1. tk¡p dhft, fd D;k fuEu f}?kkr lehdj.k gSa %
(i) (x + 1)
2
 = 2(x – 3) (ii) x
2
 – 2x = (–2) (3 – x)
(iii) (x – 2)(x + 1) = (x – 1)(x + 3) (iv) (x – 3)(2x +1) = x(x + 5)
(v) (2x – 1)(x – 3) = (x + 5)(x – 1) (vi) x
2
 + 3x + 1 = (x – 2)
2
(vii) (x + 2)
3
 = 2x (x
2
 – 1) (viii) x
3
 – 4x
2
 – x + 1 = (x – 2)
3
2. fuEu fLFkfr;ksa dks f}?kkr lehdj.kksa osQ :i esa fu:fir dhft, %
(i) ,d vk;rkdkj Hkw[kaM dk {ks=kiQy 528 m
2 
 gSA {ks=k dh yackbZ (ehVjksa esa) pkSM+kbZ osQ
nqxqus ls ,d vf/d gSA geas Hkw[kaM dh yackbZ vkSj pkSM+kbZ Kkr djuh gSA
(ii) nks Øekxr /ukRed iw.kk±dksa dk xq.kuiQy 306 gSA gesa iw.kk±dksa dks Kkr djuk gSA
(iii) jksgu dh ek¡ mlls 26 o"kZ cM+h gSA mudh vk;q (o"kks± esa) dk xq.kuiQy vc ls rhu
o"kZ i'pkr~ 360 gks tk,xhA gesa jksgu dh orZeku vk;q Kkr djuh gSA
(iv) ,d jsyxkM+h 480 km dh nwjh leku pky ls r; djrh gSA ;fn bldh pky 8 km/h de  gksrh]
rks og mlh nwjh dks r; djus esa 3 ?kaVs vf/d ysrhA gesa jsyxkM+h dh pky Kkr djuh gSA
4.3 xq.ku[kaMksa }kjk f}?kkr lehdj.k dk gy
f}?kkr lehdj.k 2x
2
 – 3x + 1 = 0 ij fopkj dhft,A ;fn ge bl lehdj.k osQ ck,¡ i{k
esa x dks 1 ls izfrLFkkfir djsa] rks gesa izkIr gksrk gS% (2 × 1
2
) – (3 × 1) + 1 = 0 = lehdj.k
dk nk¡;k i{kA ge dgrs gSa fd 1 f}?kkr lehdj.k 2x
2
 – 3x + 1 = 0 dk ,d ewy gSA bldk
;g Hkh vFkZ gS fd 1 f}?kkr cgqin 2x
2
 – 3x + 1 dk ,d 'kwU;d gSA
O;kid :i esa] ,d okLrfod la[;k a f}?kkr lehdj.k ax
2
 + bx + c = 0, a ? 0 dk
,d ewy dgykrh gS] ;fn a a
2
 + ba + c = 0 gksA ge ;g Hkh dgrs gSa fd x = a a a f}?kkr
lehdj.k dk ,d gy gS vFkok a a a f}?kkr lehdj.k dks larq"V djrk gSA è;ku nhft,
fd f}?kkr cgqin ax
2
 + bx + c osQ 'kwU;d vkSj f}?kkr lehdj.k ax
2
 + bx + c = 0
osQ ewy ,d gh gSaA
2018-19
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