Whakaoti i te x^2+x^2/1-x^3=1

$$\frac { x ^ { 2 } + x ^ { 2 } } { 1 - x ^ { 3 } } = 1$$

Whakaoti mō x

$$x=\frac{\sqrt{5}-1}{2}\approx 0.618033989$$

$$x=\frac{-\sqrt{5}-1}{2}\approx -1.618033989$$

$$x=-1$$

Ngā Upane Otinga

Tē taea kia ōrite te tāupe x ki 1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-1\right)\left(-x^{2}-x-1\right).

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

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}=\left(x-1\right)\left(-x^{2}-x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-1 ki te -x^{2}-x-1 ka whakakotahi i ngā kupu rite.

$$2x^{2}=-x^{3}+1$$

Me tāpiri te x^{3} ki ngā taha e rua.

$$2x^{2}+x^{3}=1$$

Tangohia te 1 mai i ngā taha e rua.

$$2x^{2}+x^{3}-1=0$$

Hurinahatia te whārite ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{3}+2x^{2}-1=0$$

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 -1, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±1$$

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.

$$x=-1$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te x^{3}+2x^{2}-1 ki te x+1, kia riro ko x^{2}+x-1. Whakaotihia te whārite ina ōrite te hua ki te 0.

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

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

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

Mahia ngā tātaitai.

$$x=\frac{-1±\sqrt{5}}{2}$$

Whakaotia te whārite x^{2}+x-1=0 ina he tōrunga te ±, ina he tōraro te ±.

$$x=\frac{-\sqrt{5}-1}{2} x=\frac{\sqrt{5}-1}{2}$$

Rārangitia ngā otinga katoa i kitea.

$$x=-1 x=\frac{-\sqrt{5}-1}{2} x=\frac{\sqrt{5}-1}{2}$$