Ciro tushen hadadden lamba

A cikin wannan ɗaba'ar, za mu duba yadda za ku iya ɗaukar tushen wani hadadden lamba, da kuma yadda wannan zai iya taimakawa wajen warware ma'auni huɗun waɗanda bambance-bambancen bai kai sifili ba.

Ciro tushen hadadden lamba

Tushen square

Kamar yadda muka sani, ba shi yiwuwa a dauki tushen mummunan adadi na ainihi. Amma idan yazo ga lambobi masu rikitarwa, ana iya yin wannan aikin. Bari mu gane shi.

A ce muna da lamba z = -9. Don √-9 akwai tushen guda biyu:

z₁ = √-9 = -3i

z₁ = √-9 = 3 i

Bari mu duba sakamakon da aka samu ta hanyar warware lissafin z² = -9, ba tare da mantawa da hakan ba i² = -1:

(-3i)² = (-3)² ⋅ i² = 9 ⋅ (-1) = -9

(3i)² = 3² ⋅ i² = 9 ⋅ (-1) = -9

Don haka mun tabbatar da hakan -3i ku и 3i tushen su ne √-9.

Tushen mummunan lamba yawanci ana rubuta shi kamar haka:

√-1 = ± i

√-4 = ± 2i

√-9 = ± 3i

√-16 = ± 4i da dai sauransu.

Tushen zuwa ikon n

A ce an ba mu ma'auni na fom z = ⁿ√w… Yana da n tushen (z₀, na₁, na₂,…,z_n-1), wanda za a iya lissafta ta amfani da dabarar da ke ƙasa:

|w| shine tsarin hadadden lamba w;

φ – hujjarsa

k siga ne wanda ke ɗaukar dabi'u: k = {0, 1, 2,…, n-1}.

Ƙididdigar ƙididdiga tare da hadaddun tushen tushe

Cire tushen lambar mara kyau yana canza ra'ayin da aka saba na uXNUMXbuXNUMXb. Idan mai nuna bambanci (D) kasa da sifili, to, ba za a iya samun tushen asali ba, amma ana iya wakilta su azaman lambobi masu rikitarwa.

Example

Bari mu warware ma'auni x² - 8x + 20 = 0.

Magani

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

D = b² - 4ac = 64-80 = -16

D <0, amma har yanzu muna iya ɗaukar tushen rashin nuna bambanci:

√D = √-16 = ± 4i

Yanzu za mu iya lissafin tushen:

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

Saboda haka, da equation x² - 8x + 20 = 0 yana da hadaddun tushen conjugate guda biyu:

x₁ = 4 + 2i

x₂ = 4-2i