Whakaoti i te 30/x^2-1+7-18x/x^3+1=13/x^2-x+1

$$\frac { 30 } { x ^ { 2 } - 1 } + \frac { 7 - 18 x } { x ^ { 3 } + 1 } = \frac { 13 } { x ^ { 2 } - x + 1 }$$

Whakaoti mō x

$$x=-9$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-1,x^{3}+1,x^{2}-x+1.

$$\left(x^{2}-x+1\right)\times 30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-x+1 ki te 30.

$$30x^{2}-30x+30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-1 ki te 7-18x ka whakakotahi i ngā kupu rite.

$$30x^{2}-30x+30+25x-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

Pahekotia te -30x me 25x, ka -5x.

$$30x^{2}-5x+30-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

Pahekotia te 30x^{2} me -18x^{2}, ka 12x^{2}.

$$12x^{2}-5x+30-7=\left(x^{2}-1\right)\times 13$$

Tangohia te 7 i te 30, ka 23.

$$12x^{2}-5x+23=\left(x^{2}-1\right)\times 13$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-1 ki te 13.

$$12x^{2}-5x+23=13x^{2}-13$$

Tangohia te 13x^{2} mai i ngā taha e rua.

$$12x^{2}-5x+23-13x^{2}=-13$$

Pahekotia te 12x^{2} me -13x^{2}, ka -x^{2}.

$$-x^{2}-5x+23=-13$$

Me tāpiri te 13 ki ngā taha e rua.

$$-x^{2}-5x+23+13=0$$

Tāpirihia te 23 ki te 13, ka 36.

$$-x^{2}-5x+36=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+36. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=-36=-36$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=4 b=-9$$

Tuhia anō te -x^{2}-5x+36 hei \left(-x^{2}+4x\right)+\left(-9x+36\right).

$$\left(-x^{2}+4x\right)+\left(-9x+36\right)$$

Tauwehea te x i te tuatahi me te 9 i te rōpū tuarua.

$$x\left(-x+4\right)+9\left(-x+4\right)$$

Whakatauwehea atu te kīanga pātahi -x+4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+4\right)\left(x+9\right)$$

Hei kimi otinga whārite, me whakaoti te -x+4=0 me te x+9=0.

$$x=4 x=-9$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-1,x^{3}+1,x^{2}-x+1.

$$\left(x^{2}-x+1\right)\times 30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-x+1 ki te 30.

$$30x^{2}-30x+30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-1 ki te 7-18x ka whakakotahi i ngā kupu rite.

$$30x^{2}-30x+30+25x-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

Pahekotia te -30x me 25x, ka -5x.

$$30x^{2}-5x+30-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

Pahekotia te 30x^{2} me -18x^{2}, ka 12x^{2}.

$$12x^{2}-5x+30-7=\left(x^{2}-1\right)\times 13$$

Tangohia te 7 i te 30, ka 23.

$$12x^{2}-5x+23=\left(x^{2}-1\right)\times 13$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-1 ki te 13.

$$12x^{2}-5x+23=13x^{2}-13$$

Tangohia te 13x^{2} mai i ngā taha e rua.

$$12x^{2}-5x+23-13x^{2}=-13$$

Pahekotia te 12x^{2} me -13x^{2}, ka -x^{2}.

$$-x^{2}-5x+23=-13$$

Me tāpiri te 13 ki ngā taha e rua.

$$-x^{2}-5x+23+13=0$$

Tāpirihia te 23 ki te 13, ka 36.

$$-x^{2}-5x+36=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -5 mō b, me 36 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-1\right)\times 36}}{2\left(-1\right)}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-1\right)\times 36}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-5\right)±\sqrt{25+4\times 36}}{2\left(-1\right)}$$

Whakareatia 4 ki te 36.

$$x=\frac{-\left(-5\right)±\sqrt{25+144}}{2\left(-1\right)}$$

Tāpiri 25 ki te 144.

$$x=\frac{-\left(-5\right)±\sqrt{169}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-\left(-5\right)±13}{2\left(-1\right)}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±13}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{5±13}{-2}$$

Nā, me whakaoti te whārite x=\frac{5±13}{-2} ina he tāpiri te ±. Tāpiri 5 ki te 13.

$$x=\frac{18}{-2}$$

Whakawehe 18 ki te -2.

$$x=-9$$

Nā, me whakaoti te whārite x=\frac{5±13}{-2} ina he tango te ±. Tango 13 mai i 5.

$$x=-\frac{8}{-2}$$

Whakawehe -8 ki te -2.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=-9 x=4$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-1\right)\left(x+1\right)\left(x^{2}-x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-1,x^{3}+1,x^{2}-x+1.

$$\left(x^{2}-x+1\right)\times 30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-x+1 ki te 30.

$$30x^{2}-30x+30+\left(x-1\right)\left(7-18x\right)=\left(x^{2}-1\right)\times 13$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-1 ki te 7-18x ka whakakotahi i ngā kupu rite.

$$30x^{2}-30x+30+25x-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

Pahekotia te -30x me 25x, ka -5x.

$$30x^{2}-5x+30-18x^{2}-7=\left(x^{2}-1\right)\times 13$$

Pahekotia te 30x^{2} me -18x^{2}, ka 12x^{2}.

$$12x^{2}-5x+30-7=\left(x^{2}-1\right)\times 13$$

Tangohia te 7 i te 30, ka 23.

$$12x^{2}-5x+23=\left(x^{2}-1\right)\times 13$$

Whakamahia te āhuatanga tohatoha hei whakarea te x^{2}-1 ki te 13.

$$12x^{2}-5x+23=13x^{2}-13$$

Tangohia te 13x^{2} mai i ngā taha e rua.

$$12x^{2}-5x+23-13x^{2}=-13$$

Pahekotia te 12x^{2} me -13x^{2}, ka -x^{2}.

$$-x^{2}-5x+23=-13$$

Tangohia te 23 mai i ngā taha e rua.

$$-x^{2}-5x=-13-23$$

Tangohia te 23 i te -13, ka -36.

$$-x^{2}-5x=-36$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-5x}{-1}=-\frac{36}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{5}{-1}\right)x=-\frac{36}{-1}$$

Whakawehe -5 ki te -1.

$$x^{2}+5x=-\frac{36}{-1}$$

Whakawehe -36 ki te -1.

$$x^{2}+5x=36$$

Whakawehea te 5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{2}. Nā, tāpiria te pūrua o te \frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=36+\left(\frac{5}{2}\right)^{2}$$

Pūruatia \frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+5x+\frac{25}{4}=36+\frac{25}{4}$$

Tāpiri 36 ki te \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{169}{4}$$

Tauwehea te x^{2}+5x+\frac{25}{4}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{169}{4}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Whakarūnātia.

$$x+\frac{5}{2}=\frac{13}{2} x+\frac{5}{2}=-\frac{13}{2}$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$x=4 x=-9$$