Whakaoti i te x+1/3x-1=1-2x+1/4

$$\frac { x + 1 } { 3 x - 1 } = 1 - \frac { 2 x + 1 } { 4 }$$

Whakaoti mō x (complex solution)

$$x=\frac{7+\sqrt{119}i}{12}\approx 0.583333333+0.909059343i$$

$$x=\frac{-\sqrt{119}i+7}{12}\approx 0.583333333-0.909059343i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki \frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(3x-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 3x-1,4.

$$4\left(x+1\right)=4\left(3x-1\right)-\left(3x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x+1.

$$4x+4=4\left(3x-1\right)-\left(3x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 3x-1.

$$4x+4=12x-4-\left(3x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-1 ki te 2x+1 ka whakakotahi i ngā kupu rite.

$$4x+4=12x-4-\left(6x^{2}+x-1\right)$$

Hei kimi i te tauaro o 6x^{2}+x-1, kimihia te tauaro o ia taurangi.

$$4x+4=12x-4-6x^{2}-x+1$$

Pahekotia te 12x me -x, ka 11x.

$$4x+4=11x-4-6x^{2}+1$$

Tāpirihia te -4 ki te 1, ka -3.

$$4x+4=11x-3-6x^{2}$$

Tangohia te 11x mai i ngā taha e rua.

$$4x+4-11x=-3-6x^{2}$$

Pahekotia te 4x me -11x, ka -7x.

$$-7x+4=-3-6x^{2}$$

Tangohia te -3 mai i ngā taha e rua.

$$-7x+4-\left(-3\right)=-6x^{2}$$

Ko te tauaro o -3 ko 3.

$$-7x+4+3=-6x^{2}$$

Me tāpiri te 6x^{2} ki ngā taha e rua.

$$-7x+4+3+6x^{2}=0$$

Tāpirihia te 4 ki te 3, ka 7.

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

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.

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

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

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

Pūrua -7.

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

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-7\right)±\sqrt{49-24\times 7}}{2\times 6}$$

Whakareatia -24 ki te 7.

$$x=\frac{-\left(-7\right)±\sqrt{49-168}}{2\times 6}$$

Tāpiri 49 ki te -168.

$$x=\frac{-\left(-7\right)±\sqrt{-119}}{2\times 6}$$

Tuhia te pūtakerua o te -119.

$$x=\frac{-\left(-7\right)±\sqrt{119}i}{2\times 6}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±\sqrt{119}i}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{7±\sqrt{119}i}{12}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{119}i}{12} ina he tāpiri te ±. Tāpiri 7 ki te i\sqrt{119}.

$$x=\frac{7+\sqrt{119}i}{12}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{119}i}{12} ina he tango te ±. Tango i\sqrt{119} mai i 7.

$$x=\frac{-\sqrt{119}i+7}{12}$$

Kua oti te whārite te whakatau.

$$x=\frac{7+\sqrt{119}i}{12} x=\frac{-\sqrt{119}i+7}{12}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki \frac{1}{3} nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 4\left(3x-1\right), arā, te tauraro pātahi he tino iti rawa te kitea o 3x-1,4.

$$4\left(x+1\right)=4\left(3x-1\right)-\left(3x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x+1.

$$4x+4=4\left(3x-1\right)-\left(3x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te 3x-1.

$$4x+4=12x-4-\left(3x-1\right)\left(2x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 3x-1 ki te 2x+1 ka whakakotahi i ngā kupu rite.

$$4x+4=12x-4-\left(6x^{2}+x-1\right)$$

Hei kimi i te tauaro o 6x^{2}+x-1, kimihia te tauaro o ia taurangi.

$$4x+4=12x-4-6x^{2}-x+1$$

Pahekotia te 12x me -x, ka 11x.

$$4x+4=11x-4-6x^{2}+1$$

Tāpirihia te -4 ki te 1, ka -3.

$$4x+4=11x-3-6x^{2}$$

Tangohia te 11x mai i ngā taha e rua.

$$4x+4-11x=-3-6x^{2}$$

Pahekotia te 4x me -11x, ka -7x.

$$-7x+4=-3-6x^{2}$$

Me tāpiri te 6x^{2} ki ngā taha e rua.

$$-7x+4+6x^{2}=-3$$

Tangohia te 4 mai i ngā taha e rua.

$$-7x+6x^{2}=-3-4$$

Tangohia te 4 i te -3, ka -7.

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

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.

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

Whakawehea ngā taha e rua ki te 6.

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

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

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

Whakawehea te -\frac{7}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{12}. Nā, tāpiria te pūrua o te -\frac{7}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{6}x+\left(-\frac{7}{12}\right)^{2}=-\frac{7}{6}+\left(-\frac{7}{12}\right)^{2}$$

Pūruatia -\frac{7}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=-\frac{7}{6}+\frac{49}{144}$$

Tāpiri -\frac{7}{6} ki te \frac{49}{144} 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.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=-\frac{119}{144}$$

Tauwehea x^{2}-\frac{7}{6}x+\frac{49}{144}. 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}.

$$\left(x-\frac{7}{12}\right)^{2}=-\frac{119}{144}$$

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

$$\sqrt{\left(x-\frac{7}{12}\right)^{2}}=\sqrt{-\frac{119}{144}}$$

Whakarūnātia.

$$x-\frac{7}{12}=\frac{\sqrt{119}i}{12} x-\frac{7}{12}=-\frac{\sqrt{119}i}{12}$$

Me tāpiri \frac{7}{12} ki ngā taha e rua o te whārite.

$$x=\frac{7+\sqrt{119}i}{12} x=\frac{-\sqrt{119}i+7}{12}$$