Extracting lub hauv paus ntawm ib tug complex tooj

Hauv kev tshaj tawm no, peb yuav saib yuav ua li cas koj tuaj yeem coj lub hauv paus ntawm tus lej nyuaj, thiab qhov no yuav pab tau li cas hauv kev daws cov kev sib npaug plaub npaug uas nws qhov kev ntxub ntxaug tsawg dua xoom.

Extracting lub hauv paus ntawm ib tug complex tooj

Square cag

Raws li peb paub, nws yog tsis yooj yim sua kom coj lub hauv paus ntawm tus naj npawb tiag tiag tsis zoo. Tab sis thaum nws los txog rau cov lej complex, qhov kev ua no tuaj yeem ua tau. Cia peb xav txog nws.

Wb hais tias peb muaj tus lej z = -9. rau √-9 muaj ob lub hauv paus:

z₁ = √-9 = -3 ib

z₁ = √-9 = 3 ib

Cia peb tshawb xyuas cov txiaj ntsig tau los ntawm kev daws qhov sib npaug z² = -9, tsis nco qab lawm i² = -1:

(-3i)² = (-3)² ⋅ ib² = 9 ⋅ (-1) = -9

(3i)² = 3² ⋅ ib² = 9 ⋅ (-1) = -9

Yog li, peb tau ua pov thawj tias -3 ib и 3i yog cag √-9.

Lub hauv paus ntawm tus lej tsis zoo feem ntau yog sau li no:

√-1 = ±i

√-4 = ± 2i

√-9 = ± 3i

√-16 = ± 4i thiab lwm yam

Root rau lub zog ntawm n

Piv txwv tias peb muab qhov sib npaug ntawm daim ntawv z = ⁿ√w… Nws muaj n keeb kwm (z₀, ntawm₁, ntawm₂,..., z_n-1), uas tuaj yeem suav nrog cov qauv hauv qab no:

|w| yog lub module ntawm ib tug complex tooj w;

φ - nws qhov kev sib cav

k yog ib qho parameter uas siv cov nqi: k = {0, 1, 2,…, n-1}.

Quadratic sib npaug nrog cov hauv paus hniav

Extracting lub hauv paus ntawm tus lej tsis zoo hloov lub tswv yim ib txwm ua ntawm uXNUMXbuXNUMXb. Yog hais tias tus discriminant (D) tsawg dua xoom, ces tsis tuaj yeem muaj cov hauv paus hniav tiag tiag, tab sis lawv tuaj yeem sawv cev ua tus lej nyuaj.

Piv txwv li

Cia peb daws qhov sib npaug x² 8x + 20 = 0.

tshuaj

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

D = b² – 4 ib = 64-80 = 16

D <0, tab sis peb tseem tuaj yeem coj lub hauv paus ntawm kev ntxub ntxaug tsis zoo:

√D = √-16 = ± 4i

Tam sim no peb tuaj yeem xam cov hauv paus hniav:

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

Yog li ntawd, qhov sib npaug x² 8x + 20 = 0 Nws muaj ob tug complex conjugate keeb kwm:

x₁ = 4+2i

x₂ = 4-2 ib