7.5 P zsAz U Pgt rĪz (Graphical method of solving Quadratic equations):

 

y=mx+c (m v c gAPU) gƥz Pgtz P JAiĪz UU Pwzê.

F Pgtz PAi MAz ggSAiiVgvz JAz Pwzê.

FU ax2 +bx+ c =0 gƥz U Pgt P zsz rĪ PĪ wAiĪ.

EAv Pgt Jgq zsz rz.

 

zs 1:

ax2 +bx+ c = 0 Jߪz ax2= -bx-c JAz gAiħz. C Jgq y U īg. DU U y = ax2 v y =-bx-c

JAz Jgq PgtU Uv۪. JgqP P ga. CU MAzƣAz PrAiĪ AzU ax2 +bx+ c = 0 U Pgtz ƮU. 7.5 ĸ1

F zsz rVz.

 

zs 2:

ax2 +bx+ c U P ga. Dz x CP J PrAiĪz(y=0) CAi x ɯAi ƮU. 7.5 ĸ 2 F zsz rVz.

7.5 ĸ1: y=2x2 v y= 3+x PgtU P Jz 2x2-x-3=0 Pgt rj v g ɯ PAqĻrj.

 

jg:(zs 1)

1. 2x2-x-3=0 Pgt rĪz.

Av 1: x Pɮ ɯUU y (=2x2) Ai ɯU n֪ir.

x

0

1

-1

2

-2

y

0

2

2

8

8

(x, y)

(0,0)

(1,2)

(-1,2)

(2,8)

(-2,8)

Av 2: P Aiİ (x, y) zAPU AzU Ugw. F J AzU MAz AiĪz gSɬAz er.F AiĪz gSAiģ gAi(parabola) Jߪg.

Av 3: x 2 ɯUU y (=3+x) Ai ɯU n֪ir.

x

-1

1

y

2

4

(x, y)

(-1,2)

(1,4)

E x g 2 ɯUU zAPU qP ýzP? (y=3+x JAz y=mx+c gƥz Pgt. Ezg P MAz ggSAiiVgvz. MAz ggS JAiĮ 2 AzU P!)

Av 4: P Aiİ (x,y) zAPU AzU Ugw.F Jgq AzU er (Pzg Ȣ޸z)

F ggS v gAiU 2 AzU bâv۪. D AzU zAPU (-1, 2) v (1.5, 4.5). EU x ɯU: -1 v 1.5. -1 v 3/2 F ɯU 2x2-x-3=0 PgtP CAiĪUvz. Pgtz ƮU: -1 v 1.5.

 

v:

2x2-x-3=0 Pgt ax2 +bx+ c =0 gƥzz.

F Pgtz ƮU: x = [-b (b2-4ac)]/2a

E a=2, b= -1,c= -3

(b2-4ac) = (1+24)= 5

ƮU: (15)/4 = -1 v 3/2 P PzAi E A.

 

2. g ɯ PAqĻrAiĪz:-

FU x =DzU y Ai ɯ J?

y=2x2. gAi r. x = Dzg y=2x2= 6

PAiİ y=6 DzU, x U Jgq ɯU (C gAiz ï vPgz y CPz Jgq Pq UgwVz)

C AzU J x CP PrAiĪz(ð tz Jz A) Cz ɯU qv۪.

 

7.5 ĸ 2: 2x2+3x-5=0 F Pgt P zsAz r.

 

jg: (zs 2)

Av1: x Pɮ ɯUU y(=2x2+3x-5) Ai ɯU n֪ir:

x

0

1

-1

2

-2

-3

y

-5

0

-6

9

-3

4

(x, y)

(0,-5)

(1,0)

(-1,-6)

(2,9)

(-2,-3)

(-3,4)

 

Av 2: P Aiİ (x, y) zAPU AzU Ugw. F J AzU MAz AiĪz gSɬAz er. F AiĪz gSAi gAi.

VU PVgĪz 2x2+3x-5=0 EgĪ Az (CAzg y=0 DzU)

gAiĪ x CP (x CPz ð Aiiz Azī y=0) 2 AzU bâvz

[x= -2.5(= -5/2) v x=1]

(C gAiz ï vPgz y CPz Jgq Pq UgwVz)

1 v -5/2 EU zv Pgtz ƮUV.

v:

zv Pgt: 2x2+3x-5=0. Ez ax2 +bx+ c =0 gƥzz.

ƮU: x = [-b (b2-4ac)]/2a E a=2, b= 3,c= -5

b2-4ac = 9+40=49 (b2-4ac) = 7

ƮU: (-37)/4 = 1 v -5/2 PAiİ A.

 

 

 

 

7.5 Pv SAU

 

 

A.

Pv CAU

1

P Jz U PgtU rĪz.