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ECUACIONES

TERCERO A-B-C DE

SECUNDARIA

PROFESORA: ROXANA.E. Fe y Alegría N° 11

COLLIQUE 3ra. ZONA UGEL N° 04 – COMAS

FORMAGENERAL:

Ecuación Cuadrática I

B. Factor Común

ax2 + x + c ! " # a

$ "

E%&m'(o:

E%&m'(o):

* x2 ,x !" x-x ,! "

/.R&)u&(0&: x2 2x /1 ! "

2x2 + x + / ! " # x2 + 2 !"

REOL3CI4N 5E LA EC3A CI4N C3A5R67ICA8 9or Factoriación

A. A)'a im'(&

E%&m'(o):

* x2 ,x /2 !" x ;

x +2

-x ;-x + 2 !" x

/! ; # x

2

! 22 ra<c&) di=&r&nt&)

C.. ! >;? 2@

2 )o(ucion&)

* ,x2 ,x + / !"

2x /

2x /

-2x /-2x / ! "x

/! /2 # x

2! /2

2 ra<c&) iBua(&)

C.. ! >/2@

/ )o(ución

O)&r0a u& )& Da uti(iado(a =actoriación 'or a)'a)im'(&.

x ! " E x ,! " x

/! " #

x2

! ,

2 ra<c&)

C.. ! >"? ,@

2 )o(ucion&)

* x2 + 2x !" x-x + 2! "

x ! " E x + 2! " x

/! " # x +

2 ! "x

2! 2

C.. ! >2? "@

C. 5i=&r&ncia d&Cuadrado)

E%&m'(o):

* x2 ! ;x2 ; ! "-x + ;-x ; ! "x

/! ; # x

2! ;

C.. ! >;? ;@

* 2x2 GH ! "2-x2 ,G ! "2-x + -x ! "

x/ ! # x2 ! C.. ! >? @

R&)o(ución:

x2 2x /1 !" x 1x +

-x 1-x + ! "x

/! 1 x

2!

2. R&)u&(0&:x2 ! n2

R&)o(ución:

x2 n2 !" -x n-x + n !"x n ! " E x + n

! "

x/

! n x2

! n

. R&)u&(0&:x2 2x + / !

J2

R&)o(ución:

x2 2x + / !J2

-x /2 J2 ! "-x / J-x / + J

! "x / J ! " E x / +

J ! "

x/

! / + J # x2

! /

J

,. R&)u&(0&:x2 -;a + x + ;a

! "



1

G x2 + Gx ! "

a >G@ >G#G@ c >G@d ># @& >"# G@

/" x2 + ,x ! "

/,

; a >"# ,@ d >

,# 1@

/1

># ,@ & >2# 2@ c>,@

8 R&)u&(0& (a))iBui&nt&) &cuacion&) K )&a(a&( con%unto )o(ución.

/ x2 G ! "

a >@ ># @ c>,@d ># 1@ & >2#2@

2 x2 21! "

a >1@ >1# 1@ c>,@

d >,# 1@ & >/"#/"@

x2 /;! "

a >@ ># @ c>,@d ># 1@ & >2#2@

, ,x2 G! "

a >2@ d >2#12@ >2# 2@& >/#/@c >2@

1 Gx2 / ! "

a >/@ d >#@

>/@ & >/#/@c >/? /@

; x2 x ! "

a >@ >@ c

>/@ d ># "@ & >2@ x2 1x ! "

a >1@ >1# "@c >"# 1@ d >1@

& ># 1@

H x2 + ,x ! "

a >2@ >2# 2@c >,@d >,# 2@ & >,#"@

G x2 + Gx ! "

a >G@ >G# G@c >G@ d ># @& >"# G@



/" x2 + ,x ! "

a >"# ,@ d >,# 1@ ># ,@ c >2# 2@ &>,@

// x2 + 1x + , ! "

a >/# ,@ d >/# ,@ >,# /@& >2# @ c >/# ,@

/2 x2 + Hx + /1 ! "

a ># 1@ d ># 1@ ># @& ># 1@ c >,@

/ x2 1x + ; ! "

a ># 2@ d># 1@ ># 2@& >2# 2@ c >2# @

/, x2 Gx + 2" ! "a >,# 1@ d >,# 1@ >,# 1@& >/# H@ c >,#1@

/1 M 2 Hx + /; ! "

a >@ ># @ c>,@d ># 1@ & >2# 2@

//; x2 2 ! "

a >@ ># @ c >,@ d ># 1@ &>2# 2@

/ x2 x ! "

a >"@ >/@ c >"#/@

d >/# 2@ &>2@

/;/H 2x2 ,x ! ;

a x ! / E x! x ! E

x ! H c x ! G

d x ! /2 E x !/ & N.A.

/G x2 + Hx ! 1x2 x

a x ! 2 E x ! G x ! " E x ! G2c x ! H E x ! d x ! E x ! 1& N.A.

2" 2x2 ! 2

a N 1 N , c N d N ; & N 2



2/ x2 /x + ," ! "

a >,# ;@ ># @c >1#H@d >2# G@& >2# @

" R&)u&(0&: x 2

+x

x !1 x 22

R&)u&(0&:x2 x + ,G, ! "

a >2# 2@ d

>,G,@

22 1x2 Gx 2 ! "

a >/# ;@ d>21# /@ >/# @& >2# /1@ c >2# 2@

2 1x2 + ,x / ! "

a >/@ d>/# /1@ >/# /1@& >/# /1@ c >/1@

2, ,x2

2/x + 1 ! "a >/,# 1@ d>/# 1@ >1# /,@ &>/,# 1@c >,#1@

21 x2 ! 1"x

a ># @ d>/# @ >/# @& >/# /@ c >/#@

2; ;-x2 2 ! -x2 + /

a >2@ >/# /@c >,@d >,# ,@ & >2# 2@

2 1-2x2 + / ! -,x2 + x &indica un 0a(or &nt&ro d&(con%unto )o(ución.

a >2# @ d >2# 1@ >2# ,@ & >/# 1@c >,# 1@

Ni0&( III

/ R&)u&(0& K )&a(a una

)o(ución:2x2 Hx /" !

"

a / " c /d 2 &

2 R&)u&(0&:x2 ,x + , !

"

a >2# 2@ >2@c >2@

d >,@ & >,@

R&)u&(0&:x2 Hx + /; !

"

a >,@ >,@ c >,#,@d >2@ & >2# 2@

, R&)u&(0&:x2 /"x + 21 !

"

a >1@ >1# 1@ c >1@d >/"@ & >/"@

>2@ & >,G,@c >2@

H R&)u&(0&:-x + 2 ;x !

21

a >,@ > c >,#d >2@ & >

G R&)u&(0&:/"" ;x2 !

"

a >1,@ d >;@ >1,@ & >/"";@c >1,# 1,@

," R&)u&(0&:x2K2 H/ !

"

a >G# G@ >G@ c>G@d >H/@ & >H/# H/@

,/ R&)u&(0&:21 x2 !

"

a >1# 1@ >1@ c>1@d >21@ & >21@

,2 R&)u&(0&:/;x2 , !

"

a >/2@ d >2# 2@ >/2@ & >2@c >/2# /2@

2H x2 + 1x ! -x2 + x

a >"# /2@ d>"# @ >"# @& >"# 2@ c >/2#"@

1 R&)u&(0&:

x

2

2x H ! "a >2# ,@ d >,# ,@ >,# 2@ & >,@c >2# 2@

2G R&)u&(0&: x + ! x + /

a 12 cd 1 & /



a ># 2@ ># 2@c >2@d >@ & >2# 2@

; R&)u&(0&:Gx2 + /2x + , ! "

a >2# >2@ c >@d 2 & 2



,

R&)u&(0&:a2x2 !

2

,H

R&)u&(0&:2x2 Hx !

"

,G R&)u&(0&:;x2 /2x ! "

a >a# a@ d>a@ >a# a@& >a@ c >a@ a >"# ,@ >,# "@ c>"@

d >,@ & >2# 2@

a >;@ >;@ c >"# ;@d >;# "@ & >"# 2@

,,

R&)u&(0&:x2 ! /

a >/@ > c >/#d >"@ & >2@

,1

R&)u&(0&:x2 ;x ! "

a >"# ;@ >;# "@ c>;@d >;@ & >"@

,; R&)u&(0&:,a2x2 ,xa !

"

a >"# a@ >a@ c>"@d >,a@& >,a@

, R&)u&(0&:x2 ,J2 !

"

a >2J@ d >,J#,J@ >"# 2J@ & >"@c >2J# 2J@

MuDammmad in Mu)a a( DParimi?un mi&mro d&( BaKa( a( QiJma? =u& &(autor d&0ario)tratado))or&a)tronom<aKmat&mática? &ntr& &((o) uno d& (o) 'rim&ro)tratado) i)(ámico) ac&rca d&( á(&ra.Fu& racia) a (a traducción a( (at<n d&)u (iro ac&rca d&( )i)t&ma d&num&ración Dindú? A(oritDmi d&num&ro indorum? u& Euro'aOccid&nta( conoció &)& no0&do)o)i)t&ma d& num&ración. u ora má)im'ortant&? )in &maro? =u& )u tratadod& á(&ra u&? con &( t<tu(o (i)aa( ara Pa( muaa(a -(a ci&ncia d& (a r&ducción K

con=rontación'roa(&m&nt& )iniScaa (a ci&ncia d& (a)&cuacion&).

E( 6(B&ra d& MuDammad conti&n& in)truccion&) 'ráctica)'ara r&)o(0&r ci&rta) &cuacion&) (in&a(&) K cuadrática). TLou& (a &nt& ui&r&? dic& &( autor? cuando r&a(ia )u)cá(cu(o)U? &) un núm&roV. E)& núm&ro no &) má) u& (a)o(ución d& una &cuación.

Otro im'ortant& a(B&ri)ta ára& =u& Omar DaKKam -/",H ///? m&%or conocido &n Occid&nt& 'or )u RuaiKat? unaco(&cción d& uno);"" 'o&ma). Fu& W( &( 'rim&ro &n Dac&r una c(a)iScación)i)t&mática d& (a) &cuacion&) cúica) K r&)o(0&r a(Buna) d&&((a).

La contriución d& (o) a(&ri)ta) i)(ámico) d& (o) )i(o) MI KMII &n &( d&)arro((o d&( á(&ra Dar<a )ido má) notoria )i noDui&ra tardado tanto &n &%&rc&r )u inXu&ncia &n Euro'a?dond&? un 'oco d&)'uW)? &( á(B&ra Dar<a d& con)o(idar)&d&Sniti0am&nt&.