Whakaoti i te =x^6+12x^3=40

$$= x ^ { 6 } + 12 x ^ { 3 } = 40$$

Whakaoti mō x (complex solution)

$$x\in \sqrt[3]{2\left(\sqrt{19}+3\right)}e^{\frac{\pi i}{3}},\sqrt[3]{2\left(\sqrt{19}+3\right)}e^{\frac{5\pi i}{3}},-\sqrt[3]{2\left(\sqrt{19}+3\right)},\sqrt[3]{2\left(\sqrt{19}-3\right)}e^{\frac{4\pi i}{3}},\sqrt[3]{2\left(\sqrt{19}-3\right)},\sqrt[3]{2\left(\sqrt{19}-3\right)}e^{\frac{2\pi i}{3}}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 40 mai i ngā taha e rua.

$$x^{6}+12x^{3}-40=0$$

Whakakapia te t mō te x^{3}.

$$t^{2}+12t-40=0$$

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 12 mō te b, me te -40 mō te c i te ture pūrua.

$$t=\frac{-12±\sqrt{12^{2}-4\times 1\left(-40\right)}}{2}$$

Mahia ngā tātaitai.

$$t=\frac{-12±4\sqrt{19}}{2}$$

Whakaotia te whārite t=\frac{-12±4\sqrt{19}}{2} ina he tōrunga te ±, ina he tōraro te ±.

$$t=2\sqrt{19}-6 t=-2\sqrt{19}-6$$

Mai i te x=t^{3}, ka taea ngā otinga mā te whakaoti te whārite mō ia t.

$$x=-\sqrt[3]{2\sqrt{19}-6}e^{\frac{\pi i}{3}} x=\sqrt[3]{2\sqrt{19}-6}ie^{\frac{\pi i}{6}} x=\sqrt[3]{2\sqrt{19}-6} x=-\sqrt[3]{2\sqrt{19}+6}ie^{\frac{\pi i}{6}} x=-\sqrt[3]{2\sqrt{19}+6} x=\sqrt[3]{2\sqrt{19}+6}e^{\frac{\pi i}{3}}$$

Whakaoti mō x

$$x=\sqrt[3]{2\sqrt{19}-6}\approx 1.395529599$$

$$x=\sqrt[3]{-2\sqrt{19}-6}\approx -2.450648052$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 40 mai i ngā taha e rua.

$$x^{6}+12x^{3}-40=0$$

Whakakapia te t mō te x^{3}.

$$t^{2}+12t-40=0$$

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 12 mō te b, me te -40 mō te c i te ture pūrua.

$$t=\frac{-12±\sqrt{12^{2}-4\times 1\left(-40\right)}}{2}$$

Mahia ngā tātaitai.

$$t=\frac{-12±4\sqrt{19}}{2}$$

Whakaotia te whārite t=\frac{-12±4\sqrt{19}}{2} ina he tōrunga te ±, ina he tōraro te ±.

$$t=2\sqrt{19}-6 t=-2\sqrt{19}-6$$

I te mea ko x=t^{3}, ka riro ngā otinga mā te arotake i te x=\sqrt[3]{t} mō ia t.

$$x=\sqrt[3]{2\sqrt{19}-6} x=\sqrt[3]{-2\sqrt{19}-6}$$