Whakaoti i te x^2+3x-20/x^2+3x=8

$$x ^ { 2 } + 3 x - \frac { 20 } { x ^ { 2 } + 3 x } = 8$$

Whakaoti mō x

$$x=-2$$

$$x=-1$$

$$x=2$$

$$x=-5$$

Ngā Upane Otinga

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x\left(x+3\right).

$$x\left(x+3\right)x^{2}+3xx\left(x+3\right)-20=8x\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+3.

$$\left(x^{2}+3x\right)x^{2}+3xx\left(x+3\right)-20=8x\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}+3x ki te x^{2}.

$$x^{4}+3x^{3}+3xx\left(x+3\right)-20=8x\left(x+3\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$x^{4}+3x^{3}+3x^{2}\left(x+3\right)-20=8x\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3x^{2} ki te x+3.

$$x^{4}+3x^{3}+3x^{3}+9x^{2}-20=8x\left(x+3\right)$$

Pahekotia te 3x^{3} me 3x^{3}, ka 6x^{3}.

$$x^{4}+6x^{3}+9x^{2}-20=8x\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 8x ki te x+3.

$$x^{4}+6x^{3}+9x^{2}-20=8x^{2}+24x$$

Tangohia te 8x^{2} mai i ngā taha e rua.

$$x^{4}+6x^{3}+9x^{2}-20-8x^{2}=24x$$

Pahekotia te 9x^{2} me -8x^{2}, ka x^{2}.

$$x^{4}+6x^{3}+x^{2}-20=24x$$

Tangohia te 24x mai i ngā taha e rua.

$$x^{4}+6x^{3}+x^{2}-20-24x=0$$

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

$$x^{4}+6x^{3}+x^{2}-24x-20=0$$

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 -20, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±20,±10,±5,±4,±2,±1$$

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^{4}+6x^{3}+x^{2}-24x-20 ki te x+1, kia riro ko x^{3}+5x^{2}-4x-20. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$x^{3}+5x^{2}-4x-20=0$$

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 -20, ā, ka wehea e q te whakarea arahanga 1. Whakarārangitia ngā kaitono katoa \frac{p}{q}.

$$±20,±10,±5,±4,±2,±1$$

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.

$$x=2$$

Mā te whakatakotoranga Tauwehe, he tauwehe te x-k o te pūrau mō ia pūtake k. Whakawehea te x^{3}+5x^{2}-4x-20 ki te x-2, kia riro ko x^{2}+7x+10. Whakaotihia te whārite ina ōrite te hua ki te 0.

$$x^{2}+7x+10=0$$

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 7 mō te b, me te 10 mō te c i te ture pūrua.

$$x=\frac{-7±\sqrt{7^{2}-4\times 1\times 10}}{2}$$

Mahia ngā tātaitai.

$$x=\frac{-7±3}{2}$$

Whakaotia te whārite x^{2}+7x+10=0 ina he tōrunga te ±, ina he tōraro te ±.

$$x=-5 x=-2$$

Rārangitia ngā otinga katoa i kitea.

$$x=-1 x=2 x=-5 x=-2$$