Wehe i ke kumu o ka helu paʻakikī

Ma kēia paʻi ʻana, e nānā mākou pehea e hiki ai iā ʻoe ke lawe i ke kumu o kahi helu paʻakikī, a pehea hoʻi e hiki ai i kēia ke kōkua i ka hoʻoholo ʻana i nā haʻihaʻi quadratic nona ka hoʻokaʻawale ʻana ma lalo o ka zero.

Wehe i ke kumu o ka helu paʻakikī

Aʻa huinahā

E like me kā mākou ʻike, ʻaʻole hiki ke lawe i ke kumu o kahi helu maoli ʻino. Akā inā pili i nā helu paʻakikī, hiki ke hana i kēia hana. E noʻonoʻo kākou.

E ʻōlelo kākou he helu z = -9. No √-9 ʻelua kumu.

z₁ = √-9 = -3i

z₁ = √-9 = 3i

E nānā kākou i nā hopena i loaʻa ma ka hoʻoholo ʻana i ka hoohalike z² =-9, poina ole ia i² =-1:

(-3i)² = (-3)² ⋅ i² = 9 ⋅ (-1) = -9

(Iza)² = 3² ⋅ i² = 9 ⋅ (-1) = -9

No laila, ua hōʻoia mākou i kēlā -3i и 3i he mau aa √-9.

Ua kākau pinepine ʻia ke kumu o ka helu ʻino penei:

√-1 = ± i

√-4 = ±2i

√-9 = ±3i

√-16 = ±4i etc.

Aʻa i ka mana o n

Inā hāʻawi ʻia mākou i nā hoohalike o ke ʻano z = ⁿ√w… Ua loaʻa n nā aʻa (z₀, o₁, o₂,…, z_n-1), hiki ke helu ʻia me ka hoʻohana ʻana i ke ʻano ma lalo nei:

|w| ʻo ia ka module o kahi helu paʻakikī w;

φ - kāna hoʻopaʻapaʻa

k he ʻāpana e lawe i nā waiwai: k = {0, 1, 2,…, n-1}.

Nā hoʻohālikelike pāhāhā me nā kumu paʻakikī

ʻO ka unuhi ʻana i ke kumu o kahi helu maikaʻi ʻole e hoʻololi i ka manaʻo maʻamau o uXNUMXbuXNUMXb. Inā hoʻokaʻawale (D) ʻaʻole emi ma mua o ka ʻole, a laila ʻaʻole hiki ke loaʻa nā aʻa maoli, akā hiki ke hōʻike ʻia ma ke ʻano he helu paʻakikī.

la'ana

E hoʻoholo kākou i ka hoohalike x² – 8x + 20 = 0.

pāʻoihana

a = 1, b = -8, c = 20

D = b² – 4ac = 64 – 80 = -16

D < 0, akā hiki iā mākou ke lawe i ke kumu o ka hoʻokae ʻino:

√D = √-16 = ±4i

I kēia manawa hiki iā mākou ke helu i nā kumu:

x_1,2 = (-b ± √D)/2a = (8 ± 4i)/2 = 4 ± 2i.

No laila, ka hoohalike x² – 8x + 20 = 0 ʻelua mau aʻa conjugate paʻakikī:

x₁ = 4 + 2i

x₂ = 4 – 2i