Whakaoti i te 1/2-x-1=1/x-2-6-x/3x^2-12

$$\frac{ 1 }{ 2-x } -1 = \frac{ 1 }{ x-2 } - \frac{ 6-x }{ 3 { x }^{ 2 } -12 }$$

Whakaoti mō x

$$x=-3$$

$$x=\frac{2}{3}\approx 0.666666667$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(x-2\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o 2-x,x-2,3x^{2}-12.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(6-x\right)$$

Whakareatia te 3 ki te -1, ka -3.

$$-6-3x-3\left(x-2\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te x-2.

$$-6-3x+\left(-3x+6\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -3x+6 ki te x+2 ka whakakotahi i ngā kupu rite.

$$-6-3x-3x^{2}+12=3x+6-\left(6-x\right)$$

Tāpirihia te -6 ki te 12, ka 6.

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

Hei kimi i te tauaro o 6-x, kimihia te tauaro o ia taurangi.

$$6-3x-3x^{2}=3x+6-6+x$$

Tangohia te 6 i te 6, ka 0.

$$6-3x-3x^{2}=3x+x$$

Pahekotia te 3x me x, ka 4x.

$$6-3x-3x^{2}=4x$$

Tangohia te 4x mai i ngā taha e rua.

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

Pahekotia te -3x me -4x, ka -7x.

$$6-7x-3x^{2}=0$$

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

$$-3x^{2}-7x+6=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -3x^{2}+ax+bx+6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=-3\times 6=-18$$

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 -18.

$$1,-18 2,-9 3,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-18=-17 2-9=-7 3-6=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=2 b=-9$$

Tuhia anō te -3x^{2}-7x+6 hei \left(-3x^{2}+2x\right)+\left(-9x+6\right).

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

Tauwehea te -x i te tuatahi me te -3 i te rōpū tuarua.

$$-x\left(3x-2\right)-3\left(3x-2\right)$$

Whakatauwehea atu te kīanga pātahi 3x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-2\right)\left(-x-3\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-2=0 me te -x-3=0.

$$x=\frac{2}{3} x=-3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(x-2\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o 2-x,x-2,3x^{2}-12.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(6-x\right)$$

Whakareatia te 3 ki te -1, ka -3.

$$-6-3x-3\left(x-2\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te x-2.

$$-6-3x+\left(-3x+6\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -3x+6 ki te x+2 ka whakakotahi i ngā kupu rite.

$$-6-3x-3x^{2}+12=3x+6-\left(6-x\right)$$

Tāpirihia te -6 ki te 12, ka 6.

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

Hei kimi i te tauaro o 6-x, kimihia te tauaro o ia taurangi.

$$6-3x-3x^{2}=3x+6-6+x$$

Tangohia te 6 i te 6, ka 0.

$$6-3x-3x^{2}=3x+x$$

Pahekotia te 3x me x, ka 4x.

$$6-3x-3x^{2}=4x$$

Tangohia te 4x mai i ngā taha e rua.

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

Pahekotia te -3x me -4x, ka -7x.

$$6-7x-3x^{2}=0$$

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

$$-3x^{2}-7x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, -7 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\left(-3\right)\times 6}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-\left(-7\right)±\sqrt{49+12\times 6}}{2\left(-3\right)}$$

Whakareatia 12 ki te 6.

$$x=\frac{-\left(-7\right)±\sqrt{49+72}}{2\left(-3\right)}$$

Tāpiri 49 ki te 72.

$$x=\frac{-\left(-7\right)±\sqrt{121}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-\left(-7\right)±11}{2\left(-3\right)}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±11}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{7±11}{-6}$$

Nā, me whakaoti te whārite x=\frac{7±11}{-6} ina he tāpiri te ±. Tāpiri 7 ki te 11.

$$x=\frac{18}{-6}$$

Whakawehe 18 ki te -6.

$$x=-3$$

Nā, me whakaoti te whārite x=\frac{7±11}{-6} ina he tango te ±. Tango 11 mai i 7.

$$x=-\frac{4}{-6}$$

Whakahekea te hautanga \frac{-4}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{2}{3}$$

Kua oti te whārite te whakatau.

$$x=-3 x=\frac{2}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(x-2\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o 2-x,x-2,3x^{2}-12.

$$-6-3x+3\left(x-2\right)\left(x+2\right)\left(-1\right)=3x+6-\left(6-x\right)$$

Whakareatia te 3 ki te -1, ka -3.

$$-6-3x-3\left(x-2\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3 ki te x-2.

$$-6-3x+\left(-3x+6\right)\left(x+2\right)=3x+6-\left(6-x\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te -3x+6 ki te x+2 ka whakakotahi i ngā kupu rite.

$$-6-3x-3x^{2}+12=3x+6-\left(6-x\right)$$

Tāpirihia te -6 ki te 12, ka 6.

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

Hei kimi i te tauaro o 6-x, kimihia te tauaro o ia taurangi.

$$6-3x-3x^{2}=3x+6-6+x$$

Tangohia te 6 i te 6, ka 0.

$$6-3x-3x^{2}=3x+x$$

Pahekotia te 3x me x, ka 4x.

$$6-3x-3x^{2}=4x$$

Tangohia te 4x mai i ngā taha e rua.

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

Pahekotia te -3x me -4x, ka -7x.

$$6-7x-3x^{2}=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-7x-3x^{2}=-6$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-3x^{2}-7x=-6$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}-7x}{-3}=-\frac{6}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\left(-\frac{7}{-3}\right)x=-\frac{6}{-3}$$

Whakawehe -7 ki te -3.

$$x^{2}+\frac{7}{3}x=-\frac{6}{-3}$$

Whakawehe -6 ki te -3.

$$x^{2}+\frac{7}{3}x=2$$

Whakawehea te \frac{7}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{6}. Nā, tāpiria te pūrua o te \frac{7}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{7}{3}x+\left(\frac{7}{6}\right)^{2}=2+\left(\frac{7}{6}\right)^{2}$$

Pūruatia \frac{7}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=2+\frac{49}{36}$$

Tāpiri 2 ki te \frac{49}{36}.

$$x^{2}+\frac{7}{3}x+\frac{49}{36}=\frac{121}{36}$$

Tauwehea x^{2}+\frac{7}{3}x+\frac{49}{36}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{6}\right)^{2}=\frac{121}{36}$$

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

$$\sqrt{\left(x+\frac{7}{6}\right)^{2}}=\sqrt{\frac{121}{36}}$$

Whakarūnātia.

$$x+\frac{7}{6}=\frac{11}{6} x+\frac{7}{6}=-\frac{11}{6}$$

Me tango \frac{7}{6} mai i ngā taha e rua o te whārite.

$$x=\frac{2}{3} x=-3$$