Ke aa o ka quadratic helu kaulike: algebraic a me ka geometric manaʻo

I hōʻailona helu huinahalike ua kapaia ka lua o ka papa helu kaulike. By helu kaulike imply i ka makemakikaʻa i, i i loko o kona haku mele 'ana o kekahi a oi IKE OLE IA. Lua-mea helu kaulike - he makemakika helu kaulike ka ma ka liʻiliʻi loa kekahi i ike ole i kaʻahā like degere. Na quadratic helu kaulike - lua-mea helu kaulike hōʻike paha, ka 'e ano like ke' Aʻohe. Wehewehe i ka helu kaulike kū pono o ka mea hookahi e ho oholo ai ka huinahalike aa o ka helu kaulike. Typical quadratic helu kaulike i loko o ka mau palapala:

W * l ^ 2 + T * l + O = 0

kahi W, T - na coefficients o na aa o ka quadratic helu kaulike;

O - noa kaʻi lau waiwai;

c. - mole o ka quadratic helu kaulike (mau ua mau cia C1 a me C2).

E like me ua oleloia, o ka pilikia o ka hoʻoponopono i ka quadratic helu kaulike - loaa i ke aa o ka quadratic helu kaulike. E huli ia mau mea, e pono e huli i kekahi discriminant:

N = T ^ 2 - 4 * W * O

Ke discriminant nā papakuhikuhi e pono ai no ka loaa pāʻoihana aa C1 a me C2:

C1 = (-T + √N) / 2 * W a me ka C2 = (-T - √N) / 2 * W

Inā ka quadratic helu kaulike o na mau palapala ololi ma ke kumu o T mai he mau waiwai, i ka helu kaulike UAIAaIN e:

W * l ^ 2 + 2 * U * c. + O = 0

A me ka nana aku e like me ka 'aoʻao o kona aʻa:

C1 = [-U + √ (U ^ 2-W * O)] / W a me ka C2 = [-U - √ (U '2-W * O)] / W

Pinepine helu kaulike i i ka iki okoa helehelena ia C_2 i i ole kaʻi lau waiwai W. Ma keia hihia, no ka luna helu kaulike i ka palapala:

c. ^ 2 + F * l + L = 0

kahi F - ololi ma ka aa ona;

L - noa ololi;

c. - mole o ka huinahalike (mau ua mau cia C1 a me C2).

Kēiaʻano o ka helu kaulike ua kapaia he quadratic helu kaulike i haawiia mai. Ka inoa "hoemi" hele mai haʻilula actuation typical quadratic helu kaulike, ina ke kaʻi lau waiwai o W kumu mai o ka waiwai o kekahi. Ma kēia hihia, na aa o ka quadratic helu kaulike.

C1 = -f / 2 + √ [(F / 2) '2-L)] a me ka C2 = -f / 2 - √ [(F / 2)' 2-L)]

Ma ka hihia o ke ahiahi, cia o ke kaʻi lau waiwai o ka F kumu aʻa e i ka pāʻoihana.

C1 = -f + √ (F ^ 2-L) C2 = -f - √ (F ^ 2-L)

Inā mākou e kamailio e pili ana i nā helu kaulike quadratic, ia mea pono e hoole i ka theorem o Vieta. Ua hākālia nō i ke kēia mau kānāwai no ka hoemi quadratic helu kaulike.

c. ^ 2 + F * l + L = 0

C1 + C2 = -f a me ka C1 * C2 = L

Ma kekahi mau quadratic helu kaulike quadratic helu kaulike aʻa i nā mau panalaau:

W * l ^ 2 + T * l + O = 0

C1 + C2 = -T / W a me ka C1 * C2 = O / W

Ano, e noonoo i ke koho ana o quadratic nā helu kaulike a me kā lākou kākoʻo pāʻoihana. A pau o ia hiki e elua, me ina he lālā o ka c_2 ua nalowale, alaila, e ole eʻahā like ka helu kaulike. nolaila:

1. W * l ^ 2 + T * l = 0 o ka quadratic helu kaulikeʻano me ka noa ololi (lālā).

Ke kākoʻo pāʻoihana mea:

W * l ^ 2 = -T * l

C1 = 0, C2 = -T / W

2. W * l ^ + 2 O = 0 o ka quadratic helu kaulikeʻano me ka lua o ka makahiki, i ka wa a ka ia modulo na aa o ka quadratic helu kaulike.

Ke kākoʻo pāʻoihana mea:

W * l ^ 2 = -O

C1 = √ (-O / W), C2 = - √ (-O / W)

A pau i kēia i hōʻailona helu. E noonoo oe i ka geometric manaʻo o ka i mai he quadratic helu kaulike. ka lua o ka papa helu kaulike i loko o ka anahonua ua ho'ākāka 'ia ma kekahi kuleana pili i parabola. muli o kona kulana pinepine ka hana mea e imi i na aa o kekahi quadratic helu kaulike no nā haumāna kula kiʻekiʻe? Mau aʻa e haawi aku i ka manaʻo o nā ana, e intersect i ka papa pakuhi (parabola) me ka i kahi kūlike o (axis) - ka papamoe. Inā, ua hoʻoholo ihola i ka quadratic helu kaulike, e hele mākou i ka irrational olelo hooholo o ka aa, alaila, ke kuʻina e ole. Inā ke kumu i kekahi physical waiwai, o ka papa nā keʻa i ka axis x ma kekahi wahi. Inā o nā aʻa mau, laila, niioaaonoaaiii, - nā heluʻai o ke kuʻina.

Ua mea waiwai 'ana i kumumanaʻo ana ma lalo o ka irrational aa imply he hopena maikaʻi waiwai ma lalo o ke kumu, i ke kumu i hoʻoholo' ia. Helu cia - i kekahi olelo a io nui. Ma ka hihia o ka loaa hoʻokahi wale nō ke aa aha i na aa o ka ia. Ke kohoʻia o ka hālawa i loko o ka Cartesian i kahi kūlike o'ōnaehana hiki no hoi ke mālama '-manao ma na coefficients o ka W aa a me T. Inā W i he maikaʻi nui, na lala elua o ka parabola i kuhikuhi keu. Inā W i ka io cia, - e kō ana. Nō hoʻi, inā ke kaʻi lau waiwai B i ka maikaʻi hoailona, kahi W mea i 'oluʻolu, ka vertex o ka papa parabola mea i loko o ka "y" mai "-" e infinity "+" infinity, "pela aku" ma ka laulā o ka hua infinity i Aʻohe. Inā T - maikaʻi waiwai, a me ka W - mea io, ma ka kekahiʻaoʻao o ka abscissa.