Whakaoti i te (1+1+alpha)^3=27

$$( 1 + 1 + \alpha ) ^ { 3 } = 27$$

Whakaoti mō α

$$\alpha =1$$

Ngā Upane Otinga

Tāpirihia te 1 ki te 1, ka 2.

$$\left(2+\alpha \right)^{3}=27$$

Whakamahia te ture huarua \left(a+b\right)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3} hei whakaroha \left(2+\alpha \right)^{3}.

$$8+12\alpha +6\alpha ^{2}+\alpha ^{3}=27$$

Tangohia te 27 mai i ngā taha e rua.

$$8+12\alpha +6\alpha ^{2}+\alpha ^{3}-27=0$$

Tangohia te 27 i te 8, ka -19.

$$-19+12\alpha +6\alpha ^{2}+\alpha ^{3}=0$$

Hurinahatia te whārite ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$\alpha ^{3}+6\alpha ^{2}+12\alpha -19=0$$

Tā te Rational Root Theorem, ko ngā pūtake whakahau katoa o tētahi pūrau kei te āhua o \frac{p}{q}, ina wehea e p te kīanga pūmau -19, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±19,±1$$

Kimihia tētahi pūtake pērā mā te whakamātau i ngā uara tau tōpū katoa, e tīmata ana i te mea iti rawa mā te uara pū. Mēnā kāore he pūtake tau tōpū e kitea, whakamātauria ngā hautanga.

$$\alpha =1$$

Mā te whakatakotoranga Tauwehe, he tauwehe te \alpha -k o te pūrau mō ia pūtake k. Whakawehea te \alpha ^{3}+6\alpha ^{2}+12\alpha -19 ki te \alpha -1, kia riro ko \alpha ^{2}+7\alpha +19. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$\alpha ^{2}+7\alpha +19=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te 1 mō te a, te 7 mō te b, me te 19 mō te c i te ture pūrua.

$$\alpha =\frac{-7±\sqrt{7^{2}-4\times 1\times 19}}{2}$$

Mahia ngā tātaitai.

$$\alpha =\frac{-7±\sqrt{-27}}{2}$$

Tā te mea e kore te pūrua o tētahi tau tōraro e tautohutia ki te āpure tūturu, kāhore he rongoā.

$$\alpha \in \emptyset$$

Rārangitia ngā otinga katoa i kitea.

$$\alpha =1$$