Whakaoti i te x^4-2x^2-1=0

$$x ^ { 4 } - 2 x ^ { 2 } - 1 = 0$$

Whakaoti mō x (complex solution)

$$x=i\sqrt{\sqrt{2}-1}\approx 0.643594253i$$

$$x=-i\sqrt{\sqrt{2}-1}\approx -0-0.643594253i$$

$$x=-\sqrt{\sqrt{2}+1}\approx -1.553773974$$

$$x=\sqrt{\sqrt{2}+1}\approx 1.553773974$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakakapia te t mō te x^{2}.

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

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 -2 mō te b, me te -1 mō te c i te ture pūrua.

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

Mahia ngā tātaitai.

$$t=\frac{2±2\sqrt{2}}{2}$$

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

$$t=\sqrt{2}+1 t=1-\sqrt{2}$$

I te mea ko x=t^{2}, ka riro ngā otinga mā te arotake i te x=±\sqrt{t} mō ia t.

$$x=-\sqrt{\sqrt{2}+1} x=\sqrt{\sqrt{2}+1} x=-i\sqrt{-\left(1-\sqrt{2}\right)} x=i\sqrt{-\left(1-\sqrt{2}\right)}$$

Whakaoti mō x

$$x=-\sqrt{\sqrt{2}+1}\approx -1.553773974$$

$$x=\sqrt{\sqrt{2}+1}\approx 1.553773974$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakakapia te t mō te x^{2}.

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

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 -2 mō te b, me te -1 mō te c i te ture pūrua.

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

Mahia ngā tātaitai.

$$t=\frac{2±2\sqrt{2}}{2}$$

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

$$t=\sqrt{2}+1 t=1-\sqrt{2}$$

I te mea ko x=t^{2}, ka riro ngā otinga mā te arotake i te x=±\sqrt{t} mō t tōrunga.

$$x=\sqrt{\sqrt{2}+1} x=-\sqrt{\sqrt{2}+1}$$