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7/21/2019 Ecuación Cuadrática I
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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
! "
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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@
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/" 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
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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 & /
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a ># 2@ ># 2@c >2@d >@ & >2# 2@
; R&)u&(0&:Gx2 + /2x + , ! "
a >2# >2@ c >@d 2 & 2
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,
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&.
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