Whakaoti i te (9/x-2=2/x-1)

$$( \frac { 9 } { x } - 2 = \frac { 2 } { x - 1 } )$$

Whakaoti mō x

$$x=3$$

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-1.

$$\left(x-1\right)\times 9+x\left(x-1\right)\left(-2\right)=x\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-1 ki te 9.

$$9x-9+x\left(x-1\right)\left(-2\right)=x\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-1.

$$9x-9+\left(x^{2}-x\right)\left(-2\right)=x\times 2$$

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

$$9x-9-2x^{2}+2x=x\times 2$$

Pahekotia te 9x me 2x, ka 11x.

$$11x-9-2x^{2}=x\times 2$$

Tangohia te x\times 2 mai i ngā taha e rua.

$$11x-9-2x^{2}-x\times 2=0$$

Pahekotia te 11x me -x\times 2, ka 9x.

$$9x-9-2x^{2}=0$$

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

$$-2x^{2}+9x-9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -2x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=9 ab=-2\left(-9\right)=18$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. 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.

$$1+18=19 2+9=11 3+6=9$$

Ko te otinga te takirua ka hoatu i te tapeke 9.

$$a=6 b=3$$

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

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

Tauwehea te 2x i te tuatahi me te -3 i te rōpū tuarua.

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

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

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

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

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-1.

$$\left(x-1\right)\times 9+x\left(x-1\right)\left(-2\right)=x\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-1 ki te 9.

$$9x-9+x\left(x-1\right)\left(-2\right)=x\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-1.

$$9x-9+\left(x^{2}-x\right)\left(-2\right)=x\times 2$$

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

$$9x-9-2x^{2}+2x=x\times 2$$

Pahekotia te 9x me 2x, ka 11x.

$$11x-9-2x^{2}=x\times 2$$

Tangohia te x\times 2 mai i ngā taha e rua.

$$11x-9-2x^{2}-x\times 2=0$$

Pahekotia te 11x me -x\times 2, ka 9x.

$$9x-9-2x^{2}=0$$

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.

$$-2x^{2}+9x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 9 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-2\right)\left(-9\right)}}{2\left(-2\right)}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\left(-2\right)\left(-9\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-9±\sqrt{81+8\left(-9\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -9.

$$x=\frac{-9±\sqrt{81-72}}{2\left(-2\right)}$$

Tāpiri 81 ki te -72.

$$x=\frac{-9±\sqrt{9}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-9±3}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-9±3}{-4}$$

Nā, me whakaoti te whārite x=\frac{-9±3}{-4} ina he tāpiri te ±. Tāpiri -9 ki te 3.

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

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

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

Nā, me whakaoti te whārite x=\frac{-9±3}{-4} ina he tango te ±. Tango 3 mai i -9.

$$x=-\frac{12}{-4}$$

Whakawehe -12 ki te -4.

$$x=3$$

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,1 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x-1.

$$\left(x-1\right)\times 9+x\left(x-1\right)\left(-2\right)=x\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-1 ki te 9.

$$9x-9+x\left(x-1\right)\left(-2\right)=x\times 2$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x-1.

$$9x-9+\left(x^{2}-x\right)\left(-2\right)=x\times 2$$

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

$$9x-9-2x^{2}+2x=x\times 2$$

Pahekotia te 9x me 2x, ka 11x.

$$11x-9-2x^{2}=x\times 2$$

Tangohia te x\times 2 mai i ngā taha e rua.

$$11x-9-2x^{2}-x\times 2=0$$

Pahekotia te 11x me -x\times 2, ka 9x.

$$9x-9-2x^{2}=0$$

Me tāpiri te 9 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$9x-2x^{2}=9$$

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.

$$-2x^{2}+9x=9$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+9x}{-2}=\frac{9}{-2}$$

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

$$x^{2}+\frac{9}{-2}x=\frac{9}{-2}$$

Whakawehe 9 ki te -2.

$$x^{2}-\frac{9}{2}x=\frac{9}{-2}$$

Whakawehe 9 ki te -2.

$$x^{2}-\frac{9}{2}x=-\frac{9}{2}$$

Whakawehea te -\frac{9}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{4}. Nā, tāpiria te pūrua o te -\frac{9}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{9}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=-\frac{9}{2}+\frac{81}{16}$$

Tāpiri -\frac{9}{2} ki te \frac{81}{16} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{9}{16}$$

Tauwehea x^{2}-\frac{9}{2}x+\frac{81}{16}. 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}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{9}{16}$$

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

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Whakarūnātia.

$$x-\frac{9}{4}=\frac{3}{4} x-\frac{9}{4}=-\frac{3}{4}$$

Me tāpiri \frac{9}{4} ki ngā taha e rua o te whārite.

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