Kuchotsa muzu wa nambala yovuta

M'bukuli, tiwona momwe mungatengere muzu wa nambala yovuta, komanso momwe izi zingathandizire kuthetsa ma quadratic equations omwe tsankho lawo ndi locheperapo ziro.

Kuchotsa muzu wa nambala yovuta

square root

Monga tikudziwira, n'zosatheka kutenga muzu wa nambala yeniyeni yeniyeni. Koma zikafika pa manambala ovuta, izi zitha kuchitika. Tiyeni tiganizire.

Tinene kuti tili ndi nambala z = -9. Kwa √-9 pali mizu iwiri:

z₁ = √-9 = -3i

z₁ = √-9 = 3 ndi

Tiyeni tiwone zotsatira zomwe tapeza pothetsa equation z² =-9, osayiwala zimenezo i² =-1:

(-3i)² = (-3)² ⋅ ndi² = 9 ⋅ (-1) = -9

(3i)² = 3² ⋅ ndi² = 9 ⋅ (-1) = -9

Motero, tatsimikizira zimenezo -3 ndi и 3i ndi mizu √-9.

Muzu wa nambala yolakwika nthawi zambiri imalembedwa motere:

√-1 = ±ndi

√-4 = ±2i

√-9 = ±3i

√-16 = ±4i etc.

Muzu ku mphamvu ya n

Tiyerekeze kuti tapatsidwa ma equation a fomu z = ⁿ√w… Zatero n mizu (z₀, wa₁, wa₂,…, z_n-1), yomwe imatha kuwerengedwa pogwiritsa ntchito njira yomwe ili pansipa:

|w | ndi gawo la nambala yovuta w;

φ - malingaliro ake

k ndi parameter yomwe imatenga ma values: k = {0, 1, 2,…, n-1}.

Ma quadratic equation okhala ndi mizu yovuta

Kuchotsa muzu wa nambala yolakwika kumasintha lingaliro lanthawi zonse la uXNUMXbuXNUMXb. Ngati watsankho (D) ndi yocheperapo kuposa ziro, ndiye sipangakhale mizu yeniyeni, koma ikhoza kuimiridwa ngati manambala ovuta.

Mwachitsanzo

Tiyeni tithetse equation x² 8x + 20 = 0.

Anakonza

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

D = b² -4ac = 64 - 80 = -16

D <0, koma titha kutengabe mizu ya tsankho loipa:

√D = √-16 = ±4i

Tsopano titha kuwerengera mizu:

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

Chifukwa chake, equation x² 8x + 20 = 0 ili ndi mizu iwiri yovuta ya conjugate:

x₁ = 4 + 2i

x₂ = 4 – 2i