kf7
11
ju{ ;dLs/0f
(Quadratic Equation)
11.0 k'g/fjnf]sg
tre
Ps lbg lgdfn] cfkm\gL lbbLnfO{ elg5g\ ‘d;“u s]xL ?k}of“ 5 t/ xh'/n] dnfO{ 5 ?k}of“ lbg' eof] eg]
d;“u 10 ?k}of“ x'g]5 .’ s] o;nfO{ ;dLs/0f agfP/ xn ug{ ;lsG5 < 5nkmn ug'{xf];\ .
/ x2 - 12x + 20 = 0 dWo] s'g ju{ ;dLs/0f xf] < s] ax2 + bx + c = 0 ;w}“ ju{
;dLs/0f x'G5 < 5nkmn ug'{xf];\ . x2 - 25 = 0 df x sf] dfg - 5, 0, 5, 25 dWo] s'g s'gn] ;Gt'ng
u5{g\ < ;dLs/0f x2 - 25 = 0 nfO{ z'b\w ju{ ;dLs/0f (Pure Quadratic Equation) elgG5 eg]
x2 - 5x + 6 = 0 nfO{ ldl>t ju{ ;dLs/0f (Adfected Quadratic Equation) elgG5 . ctM ax2 – c = 0
¿ksf] ju{ ;dLs/0f z'b\w ju{ ;dLs/0f xf] . hxf“ bf];|f] kb (bx) 5}g . ax2 + bx + c = 0 ¿ksf] ;dLs/0fnfO{
ju{ ;dLs/0fsf] ;fdfGo (General or Standard) ¿k elgG5 . ju{ ;dLs/0fdf x sf] clwstd
3ftfª\s '2' x'g] ePsfn] o;nfO{ bf];|f] l8u|L (Second Degree) ;dLs/0f klg elgG5 . ju{
;dLs/0fdf x sf b'O{ cf]6f dfgx¿ (Values) x'G5g\ . h;nfO{ ju{ ;dLs/0fsf d"nx¿ (Roots of
the Quadratic Equation) elgG5 . ju{ ;dLs/0fsf] xn ljleGg tl/sfn] ug{ ;lsG5 . oxf“ ju{
;dLs/0fsf xn ug]{ ltg cf]6f ljlwx¿ k|:t't ul/Psf] 5 M
De
ve
lop
me
nt
Ce
n
0 x2 + 3x + 5 =0
(i)
v08Ls/0f ljlwåf/f ju{ ;dLs/0fsf] xn
(ii)
ju{ k'/f u/]/ ju{ ;dLs/0fsf] xn
(iii);"q
cu
lum
Cu
rri
k|of]u u/]/ ju{ ;dLs/0fsf] xn
11.1 v08Ls/0f ljlwåf/f ju{ ;dLs/0fsf]
Equation by Factorization Method)
pbfx/0f 1
xn ug'{xf];\ :
x2 – 3x + 2 = 0
;dfwfg M oxf“ ,
x2 – 3x + 2 = 0 cyjf,
x2 – (2 + 1)x + 2 = 0
cyjf, x2 – 2x – x + 2 = 0
cyjf,
cyjf, ul0ft, sIff – (
xn (Solution of the Quadratic
x(x – 2) – 1(x – 2) = 0
(x – 2) (x – 1) = 0
103
cyjf ls, x – 2 = 0 …
ls
(i),
x – 1 = 0 … (ii)
oxf“ klxnf] v08sf] ;dLs/0faf6 x = 2 / bf;|f] v08sf] ;dLs/0faf6 x =1 x'G5 . cyf{t
x = 1 / 2 x'G5 . o;/L v08Ls/0f ljlwåf/f ju{ ;dLs/0f xn ubf{ ;a} kbx¿ afof“lt/ /flvG5
/ bfof“tkm{ s]an z"Go af“sL /flvG5 . afof“tkm{sf] aLhLo cleJo~osnfO{ cl3Nnf kf7x¿df
ul/Psf k|lj|mofx¿ ckgfP/ v08Ls/0f ul/G5 . ju{ ;dLs/0fdf ePsf] rnsf b'O{ cf]6f dfgx¿
lgsflnG5 . tL dfgx¿ wgfTds jf C0ffTds s'g} klg x'g ;S5g\ .
xn ug'{xf];\ M
tre
pbfx/0f 2
Ce
n
x2 - 5x + 6 = 0
;dfwfg M oxf“,x2 - 5x + 6 = 0
cyjf,
x2 - 3x - 2x + 6 = 0
cyjf,
x(x – 3) – 2(x – 3) = 0
cyjf,
ls, x – 3 = 0 cyjf
cyjf,
∴ dfly lbOPsf ju{ ;dLs/0fsf d"nx¿
me
nt
x–2=0
x'G5 .
De
2
/
x'g\ .
x2 + 3x + 2 = 0
x2 + 3x + 2 = 0
cu
;dfwfg M oxf“, x2 + 2x + x + 2 = 0
cyjf,
cyjf,
cyjf,
(x + 2) (x + 1) = 0
ls,
x + 2 = 0 … … … (i),
cyjf,
x + 1 = 0 … … … (ii)
;dLs/0f
rri
Cu
x(x + 2) + 1(x + 2) = 0
(ii)
af6
∴ dfly lbOPsf ju{ ;dLs/0fsf d"nx¿
–1
/
(i)
af6
x = –2
/ ;dLs/0f
104
3
lum
pbfx/0f 3
xn ug'{xf];\ M
ve
x = 3 cyjf x = 2
lop
(x – 3) (x – 2) = 0
x = –1
–2
x'G5 .
x'g\ .
ul0ft, sIff – (
pbfx/0f 4
x2 + 4x + 4 = 0
;dfwfg M oxf“,
x2 + 4x + 4 = 0
cyjf,
x2 + 2x + 2x + 4 = 0
cyjf,
x(x + 2) + 2(x + 2) = 0
cyjf,
(x + 2) (x + 2) = 0
ls,
x + 2 = 0 … … … (i) ,
cyjf,
x + 2 = 0 … … … (ii)
Ce
n
tre
xn ug'{xf];\ M
nt
;dLs/0f (i) / (ii) b'jd} f x = –2 x'G5 . t;y{ dfly lbOPsf ju{ ;dLs/0fsf d"nx¿ –2 / –2 x'g\ .
ju{ k'/f u/]/ ju{ ;dLs/0fsf] xn
(Solving Quadratic Equation by Completing Square)
me
11.2
lop
pbfx/0f 5
ve
lbOPsf] ju{ ;dLs/0fnfO{ ju{ k'/f u/]/ xn ug'{xf];\ M
Cu
rri
cu
lum
De
;dfwfg M
oxf“,x2 – 4x = –4
cyjf, x2 – 4x + 4 = 0
cyjf, x2 – 2.x.2 + 22 = 0
cyjf, (x – 2)2 = 0
x – 2 = 0 cyjf x – 2 = 0 x'G5 .
∴ x = 2 / x = 2 x'G5 .
cyf{t\ ju{ ;dLs/0f x2 – 4x = –4 sf d"nx¿
x2 – 4x = –4
2
/ 2 x'g\ .
pbfx/0f 6
ju{ k'/f u/]/ ju{ ;dLs/0fsf] xn ug'{xf];\ M x2 – 7x + 12 = 0
;dfwfg M
oxf“
cyjf,
x2 – 7x + 12 = 0
x2 – 7x = –12
2
cyjf,
ul0ft, sIff – (
2
7 7
7
x 2 -2.x. +   = −12 +  
2 2
2
[∴ b'j} tkm{
7
 
2
2
yKbf]
105
2
7
−12 × 4 + 49

x−2 =
4


2
cyjf,
7
1

x−2 = 4


cyjf,
7  1

 x − 2  = ±2 

 

[∴ –12 × 4 + 49 = –48 + 49 = 1]
x−
1
7
=± 2
2
2
[∴b'j}tkm{
ju{ x6fp“bf
ca 3gfTds lrx\g ln“bf
x−
cyjf,
x=
1+7
2
cyjf,
x=
8
2
cyjf,
lum
∴x = 3
∴ ju{ ;dLs/0f
d"nx¿
6
2
x2 – 7x + 12 = 0 sf
/
4
3
x'g\ .
rri
pbfx/0f 7
Cu
ax2 + bx + c = 0
;dfwfg M oxf“
nfO{ ju{ k'/f u/]/ xn ug'{xf];\ .
ax2 + bx + c = 0
cyjf,
ax2 + bx = –c
cyjf,
ax2 bx −c
+
=
a
a
a
cyjf,
x2 + 2.x.
cyjf,
c b2
b

x +  = − + 2
a 4a
2a 

[∴ b'j}tkm{ a n] efu ubf{ ]
2
106
x=
De
cyjf,
∴ x = 4
ju{ ;dLs/0f
1 7
x=− +
2 2
−1 + 7
x=
2
lop
1 7
+
2 2
cyjf
ve
x=
7
1
=−
2
2
me
7 1
x− =
2 2
cyjf,
]
C0ffTds lrx\g ln“bf
cu
tre
2
Ce
n
∴
[∴ a2 – 2ab + b2 = (a – b2) k|of]u ubf{ ]
nt
cyjf,
2
b b 
−c  b 
+
=
+
2a  2a 
a  2a 
2
[∴
b'j}tkm{
b
 
 2a 
2
[∴ (a + b)2 = a2 + 2ab + b2
yKbf]
k|of]u ubf{]
ul0ft, sIff – (
6
b 
−4ac + b2

x
+
=

2a 
4a2

cyjf,
2
b   b2 − 4ac

 x + 2a  = ± 2a

 
x+
b
b2 − 4ac
=±
2a
2a
cyjf,
x=
−b
b2 − 4ac
±
2a
2a
Ce
n
−b ± b2 − 4ac
2a
x=
ax2 + bx + c = 0
sf d"nx¿
−b + b2 − 4ac
2a
/
−b − b2 − 4ac
2a
lop
t;y{ ju{ ;dLs/0f
nt
∴
[b'j}tkm{ ju{ x6fp“bf]
tre
cyjf,
2




me
2
cyjf,
x=
ax2 + bx + c = 0
De
nfO{
;“u t'ngf ubf{
−b ± b2 − 4ac
2a
a = 1, b = –5
/
nfO{
c = –6
−b ± b2 − 4ac
2a
cu
;"qcg';f/,
x2 – 5x – 6 = 0
x=
lum
h:t} ju{ ;dLs/0f
x'G5 .
ve
ctM ju{ ;dLs/0f ax2 + bx + c = 0 ¿kdf /x“bf To;sf d"nx¿ lgsfNg
;"qsf] ¿kdf k|of]u ug{ ;lsG5 .
x'g\ .
− (− 5) ±
Cu
rri
oxf“ a, b / c sf dfgx¿ k|lt:yfkg ubf{
(− 5)2 − 4(− 6)(1)
x=
=
5 ± 25 + 24
2
=
5 ± 49
2
=
5±7
2
ul0ft, sIff – (
2.1
107
ca wgfTds lrx\g ln“bf
C0ffTds lrx\g ln“bf
x=
5+7
x=
2 =
12
=
2
5−7
2
−2
2
=-1
=6
11.3
sf d"nx¿
6
/
–1
;"q k|of]u u/]/ ju{ ;dLs/0fsf] xn
8
nt
pbfx/0f
x2 – 5x – 24 = 0
me
;"q k|of]u u/L lbOPsf] ju{ ;dLs/0fsf] xn ug'{xf];\ M
ca,
x=
5 ± 25 + 96
2
5 ± 121
2
5 ± 11
=
2
x=
ca, wgfTds lrx\g ln“bf
5 + 11
2
16
=
2 =
= 8
108
/
c = –24
x'G5 .
/ c sf dfgx¿ k|lt:yfkg ubf{
]
C0ffTds lrx\g ln“bf
x=
x=
∴ t;y{ ju{
a = 1, b = –5
Cu
−(−5) ± (−5)2 − 4.(−24).1
[ ∴a, b
2×1
lum
=
;"q k|of]u ubf{
;“u t'ngf ubf{
cu
−b ± b2 − 4ac
2a
ax2 + bx + c = 0
rri
=
nfO{
ve
x2 – 5x – 24 = 0
De
oxf“
lop
;dfwfg M
x'g\ .
tre
x2 – 5x – 6 = 0
Ce
n
t;y{ ju{ ;dLs/0f
5 − 11
2
−6
2
= –3
;dLs/0f x – 5x – 24 = 0 sf
2
d"nx¿
–3
/
8
x'g\ .
ul0ft, sIff – (
cEof; 11.1
tn lbOPsf ju{ ;dLs/0fx¿ xn ug'{xf];\ / hf“r]/ klg x]g'{xf];\ M
(i) (x – 1) (x + 2) = 0
ii) (x – 2) (x – 3) = 0
(iii) (x – 5) (x – 3) = 0
iv) (3x – 9) (x – 5) = 0
2.
v08Ls/0f ljlwåf/f xn ug'{xf];\ M
(i) x(2x + 1) = 3x
(ii) x(x – 3) + 4x = 0
(iii) x2 – x – 12 = 0
(iv) x2 – x – 20 = 0
(v) x2 + 11x + 30 = 0
(vi) x2 – 7x + 12 = 0
(vii)x2 – 2x – 35 = 0
(viii) x2 – 13x + 42 = 0
(ix) –x2 + 16x – 63 = 0
(x) x2 = 1225
3.
ju{ k'/f u/]/ xn ug'{xf];\ M
(i) x2 – 10x + 25 = 0
(ii) x2 – 18x + 81 = 0
(iii) x2 – 4x – 21 = 0
(iv) x2 – 4x – 45 = 0
(v) –x2 + 4x + 77 = 0
(vi) x2 + 117 = 22x
(vii)x2 + x12 = 12 (viii) x2 – 2x + 34 = 0
(ix) x2 +
4.
;"q k|of]u u/]/ xn ug'{xf];\ M
(i) x2 – 10x + 21 = 0
(iii) x2 = 2x + 143
(v) x2 + 3x = 28
(vii) x  x +  =
7  49

(ix)
ve
lop
me
nt
Ce
n
tre
1.
De
15
= 2x
16
2 x 35
=
3
9
lum
(x) x2 +
(ii) x2 – 17x + 72 = 0

Cu
rri
cu
(iv) x2 – 30x + 221 = 0
2
3
1
1
1
+
= x −2 x +3 5
ul0ft, sIff – (
(vi) x2 – 5x = 66
(viii) x2 + 2x = 323
(x)
1
1
1
+
=
x + 2 x − 3 10
109