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

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

Whakaoti mō x

$$x=3$$

$$x=\frac{1}{2}=0.5$$

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 2\left(x-1\right)\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 2x-2,x+1,x-1.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+1 ki te 3.

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

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

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

Pahekotia te 3x me 6x, ka 9x.

$$9x+3-6=\left(2x+2\right)x$$

Tangohia te 6 i te 3, ka -3.

$$9x-3=\left(2x+2\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x+2 ki te x.

$$9x-3=2x^{2}+2x$$

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

$$9x-3-2x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$9x-3-2x^{2}-2x=0$$

Pahekotia te 9x me -2x, ka 7x.

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

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

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

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

$$a+b=7 ab=-2\left(-3\right)=6$$

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 6.

$$1,6 2,3$$

Tātaihia te tapeke mō ia takirua.

$$1+6=7 2+3=5$$

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

$$a=6 b=1$$

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

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

Tauwehea te 2x i te tuatahi me te -1 i te rōpū tuarua.

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

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

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

Hei kimi otinga whārite, me whakaoti te -x+3=0 me te 2x-1=0.

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

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 2\left(x-1\right)\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 2x-2,x+1,x-1.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+1 ki te 3.

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

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

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

Pahekotia te 3x me 6x, ka 9x.

$$9x+3-6=\left(2x+2\right)x$$

Tangohia te 6 i te 3, ka -3.

$$9x-3=\left(2x+2\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x+2 ki te x.

$$9x-3=2x^{2}+2x$$

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

$$9x-3-2x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$9x-3-2x^{2}-2x=0$$

Pahekotia te 9x me -2x, ka 7x.

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

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.

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

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

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

Pūrua 7.

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

Whakareatia -4 ki te -2.

$$x=\frac{-7±\sqrt{49+8\left(-3\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -3.

$$x=\frac{-7±\sqrt{49-24}}{2\left(-2\right)}$$

Tāpiri 49 ki te -24.

$$x=\frac{-7±\sqrt{25}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-7±5}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-7±5}{-4}$$

Nā, me whakaoti te whārite x=\frac{-7±5}{-4} ina he tāpiri te ±. Tāpiri -7 ki te 5.

$$x=-\frac{2}{-4}$$

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

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

Nā, me whakaoti te whārite x=\frac{-7±5}{-4} ina he tango te ±. Tango 5 mai i -7.

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

Whakawehe -12 ki te -4.

$$x=3$$

Kua oti te whārite te whakatau.

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

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 2\left(x-1\right)\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 2x-2,x+1,x-1.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+1 ki te 3.

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

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

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

Pahekotia te 3x me 6x, ka 9x.

$$9x+3-6=\left(2x+2\right)x$$

Tangohia te 6 i te 3, ka -3.

$$9x-3=\left(2x+2\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2x+2 ki te x.

$$9x-3=2x^{2}+2x$$

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

$$9x-3-2x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$9x-3-2x^{2}-2x=0$$

Pahekotia te 9x me -2x, ka 7x.

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

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

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

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.

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

Whakawehea ngā taha e rua ki te -2.

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

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

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

Whakawehe 7 ki te -2.

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

Whakawehe 3 ki te -2.

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

Whakawehea te -\frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{4}. Nā, tāpiria te pūrua o te -\frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=-\frac{3}{2}+\left(-\frac{7}{4}\right)^{2}$$

Pūruatia -\frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=-\frac{3}{2}+\frac{49}{16}$$

Tāpiri -\frac{3}{2} ki te \frac{49}{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.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{25}{16}$$

Tauwehea te x^{2}-\frac{7}{2}x+\frac{49}{16}. 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}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{25}{16}$$

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

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Whakarūnātia.

$$x-\frac{7}{4}=\frac{5}{4} x-\frac{7}{4}=-\frac{5}{4}$$

Me tāpiri \frac{7}{4} ki ngā taha e rua o te whārite.

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