Whakaoti i te x^8-10x^4=-9

$$x ^ { 8 } - 10 x ^ { 4 } = - 9$$

Whakaoti mō x (complex solution)

$$x\in \sqrt{3},-\sqrt{3}i,-\sqrt{3},\sqrt{3}i,-1,1,i,-i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 9 ki ngā taha e rua.

$$x^{8}-10x^{4}+9=0$$

Whakakapia te t mō te x^{4}.

$$t^{2}-10t+9=0$$

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 -10 mō te b, me te 9 mō te c i te ture pūrua.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 1\times 9}}{2}$$

Mahia ngā tātaitai.

$$t=\frac{10±8}{2}$$

Whakaotia te whārite t=\frac{10±8}{2} ina he tōrunga te ±, ina he tōraro te ±.

$$t=9 t=1$$

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

$$x=-\sqrt{3}i x=\sqrt{3}i x=-\sqrt{3} x=\sqrt{3} x=-1 x=-i x=i x=1$$

Whakaoti mō x

$$x=-\sqrt{3}\approx -1.732050808$$

$$x=\sqrt{3}\approx 1.732050808$$

$$x=1$$

$$x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 9 ki ngā taha e rua.

$$x^{8}-10x^{4}+9=0$$

Whakakapia te t mō te x^{4}.

$$t^{2}-10t+9=0$$

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 -10 mō te b, me te 9 mō te c i te ture pūrua.

$$t=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 1\times 9}}{2}$$

Mahia ngā tātaitai.

$$t=\frac{10±8}{2}$$

Whakaotia te whārite t=\frac{10±8}{2} ina he tōrunga te ±, ina he tōraro te ±.

$$t=9 t=1$$

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

$$x=\sqrt{3} x=-\sqrt{3} x=1 x=-1$$