Whakaoti i te 64x^3+729=0

$$64 x ^ { 3 } + 729 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-9\sqrt{3}i+9}{8}\approx 1.125-1.948557159i$$

$$x = -\frac{9}{4} = -2\frac{1}{4} = -2.25$$

$$x=\frac{9+9\sqrt{3}i}{8}\approx 1.125+1.948557159i$$

Ngā Upane Otinga

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 729, ā, ka wehea e q te whakarea arahanga 64. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±\frac{729}{64},±\frac{729}{32},±\frac{729}{16},±\frac{729}{8},±\frac{729}{4},±\frac{729}{2},±729,±\frac{243}{64},±\frac{243}{32},±\frac{243}{16},±\frac{243}{8},±\frac{243}{4},±\frac{243}{2},±243,±\frac{81}{64},±\frac{81}{32},±\frac{81}{16},±\frac{81}{8},±\frac{81}{4},±\frac{81}{2},±81,±\frac{27}{64},±\frac{27}{32},±\frac{27}{16},±\frac{27}{8},±\frac{27}{4},±\frac{27}{2},±27,±\frac{9}{64},±\frac{9}{32},±\frac{9}{16},±\frac{9}{8},±\frac{9}{4},±\frac{9}{2},±9,±\frac{3}{64},±\frac{3}{32},±\frac{3}{16},±\frac{3}{8},±\frac{3}{4},±\frac{3}{2},±3,±\frac{1}{64},±\frac{1}{32},±\frac{1}{16},±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

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.

$$x=-\frac{9}{4}$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te 64x^{3}+729 ki te 4\left(x+\frac{9}{4}\right)=4x+9, kia riro ko 16x^{2}-36x+81. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$16x^{2}-36x+81=0$$

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 16 mō te a, te -36 mō te b, me te 81 mō te c i te ture pūrua.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 16\times 81}}{2\times 16}$$

Mahia ngā tātaitai.

$$x=\frac{36±\sqrt{-3888}}{32}$$

Whakaotia te whārite 16x^{2}-36x+81=0 ina he tōrunga te ±, ina he tōraro te ±.

$$x=\frac{-9i\sqrt{3}+9}{8} x=\frac{9+9i\sqrt{3}}{8}$$

Rārangitia ngā otinga katoa i kitea.

$$x=-\frac{9}{4} x=\frac{-9i\sqrt{3}+9}{8} x=\frac{9+9i\sqrt{3}}{8}$$

Whakaoti mō x

$$x = -\frac{9}{4} = -2\frac{1}{4} = -2.25$$

Ngā Upane Otinga

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 729, ā, ka wehea e q te whakarea arahanga 64. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±\frac{729}{64},±\frac{729}{32},±\frac{729}{16},±\frac{729}{8},±\frac{729}{4},±\frac{729}{2},±729,±\frac{243}{64},±\frac{243}{32},±\frac{243}{16},±\frac{243}{8},±\frac{243}{4},±\frac{243}{2},±243,±\frac{81}{64},±\frac{81}{32},±\frac{81}{16},±\frac{81}{8},±\frac{81}{4},±\frac{81}{2},±81,±\frac{27}{64},±\frac{27}{32},±\frac{27}{16},±\frac{27}{8},±\frac{27}{4},±\frac{27}{2},±27,±\frac{9}{64},±\frac{9}{32},±\frac{9}{16},±\frac{9}{8},±\frac{9}{4},±\frac{9}{2},±9,±\frac{3}{64},±\frac{3}{32},±\frac{3}{16},±\frac{3}{8},±\frac{3}{4},±\frac{3}{2},±3,±\frac{1}{64},±\frac{1}{32},±\frac{1}{16},±\frac{1}{8},±\frac{1}{4},±\frac{1}{2},±1$$

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.

$$x=-\frac{9}{4}$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te 64x^{3}+729 ki te 4\left(x+\frac{9}{4}\right)=4x+9, kia riro ko 16x^{2}-36x+81. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$16x^{2}-36x+81=0$$

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 16 mō te a, te -36 mō te b, me te 81 mō te c i te ture pūrua.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 16\times 81}}{2\times 16}$$

Mahia ngā tātaitai.

$$x=\frac{36±\sqrt{-3888}}{32}$$

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

$$x\in \emptyset$$

Rārangitia ngā otinga katoa i kitea.

$$x=-\frac{9}{4}$$