Whakaoti i te x^4+1/x^4=119

$${ x }^{ 4 } + \frac{ 1 }{ { x }^{ 4 } } =119$$

Whakaoti mō x (complex solution)

$$x\in \frac{-\sqrt{13}-3}{2},-\frac{\sqrt{13}i}{2}-\frac{3}{2}i,\frac{\sqrt{13}i}{2}+\frac{3}{2}i,\frac{\sqrt{13}+3}{2},-\frac{\sqrt{13}i}{2}+\frac{3}{2}i,\frac{3-\sqrt{13}}{2},\frac{\sqrt{13}i}{2}-\frac{3}{2}i,\frac{\sqrt{13}-3}{2}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x^{4}.

$$x^{4}x^{4}+1=119x^{4}$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te 4 me te 4 kia riro ai te 8.

$$x^{8}+1=119x^{4}$$

Tangohia te 119x^{4} mai i ngā taha e rua.

$$x^{8}+1-119x^{4}=0$$

Whakakapia te t mō te x^{4}.

$$t^{2}-119t+1=0$$

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 -119 mō te b, me te 1 mō te c i te ture pūrua.

$$t=\frac{-\left(-119\right)±\sqrt{\left(-119\right)^{2}-4\times 1\times 1}}{2}$$

Mahia ngā tātaitai.

$$t=\frac{119±33\sqrt{13}}{2}$$

Whakaotia te whārite t=\frac{119±33\sqrt{13}}{2} ina he tōrunga te ±, ina he tōraro te ±.

$$t=\frac{33\sqrt{13}+119}{2} t=\frac{119-33\sqrt{13}}{2}$$

Mai i te x=t^{4}, ka taea ngā otinga mā te whakaoti te whārite mō ia t.

$$x=-\left(\frac{\sqrt{13}i}{2}+\frac{3}{2}i\right) x=-\frac{\sqrt{13}+3}{2} x=\frac{\sqrt{13}i}{2}+\frac{3}{2}i x=\frac{\sqrt{13}+3}{2} x=-\frac{\sqrt{13}i}{2}+\frac{3}{2}i x=\frac{3-\sqrt{13}}{2} x=-\left(-\frac{\sqrt{13}i}{2}+\frac{3}{2}i\right) x=-\frac{3-\sqrt{13}}{2}$$

Whakaoti mō x

$$x=\frac{-\sqrt{13}-3}{2}\approx -3.302775638$$

$$x = \frac{\sqrt{13} + 3}{2} \approx 3.302775638$$

$$x=\frac{\sqrt{13}-3}{2}\approx 0.302775638$$

$$x=\frac{3-\sqrt{13}}{2}\approx -0.302775638$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x^{4}.

$$x^{4}x^{4}+1=119x^{4}$$

Hei whakarea i ngā pū o te pūtake kotahi, me tāpiri ō rātou taupū. Tāpiria te 4 me te 4 kia riro ai te 8.

$$x^{8}+1=119x^{4}$$

Tangohia te 119x^{4} mai i ngā taha e rua.

$$x^{8}+1-119x^{4}=0$$

Whakakapia te t mō te x^{4}.

$$t^{2}-119t+1=0$$

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 -119 mō te b, me te 1 mō te c i te ture pūrua.

$$t=\frac{-\left(-119\right)±\sqrt{\left(-119\right)^{2}-4\times 1\times 1}}{2}$$

Mahia ngā tātaitai.

$$t=\frac{119±33\sqrt{13}}{2}$$

Whakaotia te whārite t=\frac{119±33\sqrt{13}}{2} ina he tōrunga te ±, ina he tōraro te ±.

$$t=\frac{33\sqrt{13}+119}{2} t=\frac{119-33\sqrt{13}}{2}$$

I te mea ko x=t^{4}, ka riro ngā otinga mā te arotake i te x=±\sqrt[4]{t} mō t tōrunga.

$$x=\frac{\sqrt{13}+3}{2} x=-\frac{\sqrt{13}+3}{2} x=-\frac{3-\sqrt{13}}{2} x=\frac{3-\sqrt{13}}{2}$$