47
pirobiT, rom arSiis farTobi ar iyos fotosuraTis farTobze naklebi da gaorma-
gebul farTobze meti.
moswavleTa yuradReba gavamaxviloT imaze, rom arSiis farTobs ori pirobis
dakmayofileba moeTxoveba, TiToeuli piroba Sesabamisi algebruli TanafardobiT
aRiwereba.
wina masalasTan kavSiris konteqstSi, vimeorebT kvadratuli funqciis yofaqce-
vas, grafiks da kvadratuli utolobis amoxsnis sqemebs.
moswavleTa ZiriTadi aqtivoba gakveTilze dasmuli amocanis maTematikuri mod-
elis agebas ukavSirdeba _ TiToeuli pirobis Sesabamisad, miiReba ori kvadratuli
utoloba, romelsac arSiis siganis aRmniSvneli, x cvladi unda akmayofilebdes.
maSasadame, veZebT x cvladis yvela im mniSvnelobas, romelic orive utolobas ak-
mayofilebs. saZiebeli simravle miiReba am utolobebis amonaxsnTa simravleebis
TanakveTiT. am daskvnamde misvlis Semdeg, moswavleebi advilad pouloben sistemis
amonaxsns da ayalibeben sistemis amoxsnis wess.
amis Semdeg moswavleebis aqtivobebi dakavSirebulia konkretuli sistemebis
amoxsnebis povnasTan. amisTvis SeiZleba gamoviyenoT saxelmZRvanelos Teoriul
nawilSi warmodgenili magaliTebi.
codnis ganmtkicebis mizniT sasurvelia SevTavazoT sakontrolo kiTxvebi da
ramdenime amocana, romelTa amoxsna did dros ar moiTxovs, magaliTad, 3, 6 da
9-is g.
saSinao davalebad SeiZleba SevTavazoT 1-2, 4-5 7-10 amocanebi.
codnis ganmtkicebisTvis saWiro muSaoba Semdeg gakveTilze grZeldeba. vxsniT
saxelmZRvaneloSi warmodgenil amocanebs.
erTi gakveTili SeiZleba davuTmoT jgufur muSaobas da SevTavazoT moswav-
leebs amocanebi, romlebic kvlevas, problemis gadawyvetisas arastandartul mid-
gomas, matematikuri modelis Seqmnasa da gamoyenebas gulisxmobs (amocanebi 19, 20,
24, 25, 26).
miTiTebebi amocanebis amosaxsnelad
2
sistemis I utoloba asec Caiwereba (2x–1)
2
≥
0. cxadia, misi amonaxsnTa simravlea
(–∞;+∞). amitom sistemis amonaxsnebs warmoadgens II utoloba.
3
sistemis I utolobas amonaxsni ara aqvs, sistemasac ara aqvs amonaxsni.
4
sistemis I utolobas akmayofilebs x-is nebismieri mniSivneloba, amitom
sistemis amonaxsnTa simravle emTxveva II utolobis amonaxsnTa simravles.
5
(2x–1)
2
≤
0 utolobas akmayofilebs erTaderTi mniSvneloba _ x=12. is II utolo-
basac akmayofilebs. e. i. sistemis amonaxsnia mxolod
x=12.
6
mocemuli sistemis tolfasi sistemaa
Z
[
\
]
]
]
]
(x–2)(x–6)≥0
(x–2)(x–6)
(x–3)(x+1) ≤0
pasuxi: x∈[–1; 2].
(
x–3)(
x+1)
48
7
Z
[
\
]
]
]
]
(x–4)(x+3) ≤0
(x–4)(x+3)
(x–4)(x+4)≥0
(x–4)(x+4)
utolobebis amonaxsnTa simravleebis TanakveTa erTelementiani simravlea _ {4}.
11
x
2
+1>0 da x
2
+x+1>0 utolobebs akmayofilebs x-is nebismieri mniSvneloba,
amitom
e) sistema tolfasia
v) tolfasia
Z
[
\
]
]
]
]
x–1≤0
Z
[
\
]
]
]
]
x–2
>0
x(
x–10)
>0 sistemis,
x(x+1,2)≥0 sistemis.
13
b) mocemulis tolfasi utolobaa
Z
[
\
]
]
]
]
5(x–1,2)(x+5)≥0
(x+4)(x–1)<0
gamovsaxoT maTi amonaxsnebi geometriulad
axla cxadia, rom am sistemas amonaxsni ara aqvs (sistema araTavsebadia).
d) sistemis I utolobas ara aqvs amonaxsni, amitom sistemas ara aqvs amonaxsni.
e) sistemis I utolobas akmayofilebs x-is nebismieri mniSvneloba. amitom siste-
mis amonaxsni emTxveva II utolobis amonaxsns.
14
a) orive utolobas akmayofilebs x-is nebismieri mniSvneloba (pirvelis
D<0, meoris
D=0).
pasuxi:
(–∞; +∞).
b) I utolobas akmayofilebs x-is nemismieri mniSvneloba (D<0), amitom sistemis
amonaxsni emTxveva II utolobis amonaxsns: (3x+1)
2
>0 utolobas akmayofilebs x-is
nebismieri mniSvneloba, garda
x=– 13-isa.
pasuxi:
(
-∞;
– 13
)
∪
(
– 13; +∞
)
.
g) I utolobas _ (2x+1)
2
≤
0 akmayofilebs x-is erTaderTi mniSvneloba x= – 12, ro-
melic II utolobasac akmayofilebs.
pasuxi:
{
– 12
}
.
d) II utolobas _ (x–2)
2
≤
0 akmayofilebs x-is erTaderTi mniSvneloba x=2, romelic
I utolobas ar akmayofilebs, amitom sistemas ara aqvs amonaxsni.
e) (x–1)
2
≤
0 utolobis amonaxsnia x=1, romelic II utolobasac akmayofilebs.
pasuxi: {1}.
v) mocemulis tolfasi sistemaa:
Z
[
\
]
]
]
]
(2x–3)
2
>0
5(x–1)
(
x– 45
)
≥
0
gamovsaxoT TiToeuli utolobis amonaxsnTa
simravle ricxviT RerZze da vipovoT maTi
TanakveTa:
pasuxi:
(
-∞; 45
)
∪
[
1; 32
)
∪
(
3
2; +∞
)
.
4