Whakaoti i te 4/x-2-x-3/x-4=0

$$\frac{ 4 }{ x-2 } - \frac{ x-3 }{ x-4 } =0$$

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{7}i+9}{2}\approx 4.5-1.322875656i$$

$$x=\frac{9+\sqrt{7}i}{2}\approx 4.5+1.322875656i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 2,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-4\right)\left(x-2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-2,x-4.

$$\left(x-4\right)\times 4-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x-16-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x-16-\left(x^{2}-5x+6\right)=0$$

Hei kimi i te tauaro o x^{2}-5x+6, kimihia te tauaro o ia taurangi.

$$4x-16-x^{2}+5x-6=0$$

Pahekotia te 4x me 5x, ka 9x.

$$9x-16-x^{2}-6=0$$

Tangohia te 6 i te -16, ka -22.

$$9x-22-x^{2}=0$$

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.

$$-x^{2}+9x-22=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 9 mō b, me -22 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-22\right)}}{2\left(-1\right)}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\left(-22\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-9±\sqrt{81+4\left(-22\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -22.

$$x=\frac{-9±\sqrt{81-88}}{2\left(-1\right)}$$

Tāpiri 81 ki te -88.

$$x=\frac{-9±\sqrt{-7}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -7.

$$x=\frac{-9±\sqrt{7}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-9±\sqrt{7}i}{-2}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{7}i}{-2} ina he tāpiri te ±. Tāpiri -9 ki te i\sqrt{7}.

$$x=\frac{-9+\sqrt{7}i}{-2}$$

Whakawehe -9+i\sqrt{7} ki te -2.

$$x=\frac{-\sqrt{7}i+9}{2}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{7}i}{-2} ina he tango te ±. Tango i\sqrt{7} mai i -9.

$$x=\frac{-\sqrt{7}i-9}{-2}$$

Whakawehe -9-i\sqrt{7} ki te -2.

$$x=\frac{9+\sqrt{7}i}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{7}i+9}{2} x=\frac{9+\sqrt{7}i}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 2,4 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-4\right)\left(x-2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-2,x-4.

$$\left(x-4\right)\times 4-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x-16-\left(x-2\right)\left(x-3\right)=0$$

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

$$4x-16-\left(x^{2}-5x+6\right)=0$$

Hei kimi i te tauaro o x^{2}-5x+6, kimihia te tauaro o ia taurangi.

$$4x-16-x^{2}+5x-6=0$$

Pahekotia te 4x me 5x, ka 9x.

$$9x-16-x^{2}-6=0$$

Tangohia te 6 i te -16, ka -22.

$$9x-22-x^{2}=0$$

Me tāpiri te 22 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$9x-x^{2}=22$$

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.

$$-x^{2}+9x=22$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+9x}{-1}=\frac{22}{-1}$$

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

$$x^{2}+\frac{9}{-1}x=\frac{22}{-1}$$

Whakawehe 9 ki te -1.

$$x^{2}-9x=\frac{22}{-1}$$

Whakawehe 22 ki te -1.

$$x^{2}-9x=-22$$

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

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-22+\left(-\frac{9}{2}\right)^{2}$$

Pūruatia -\frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-9x+\frac{81}{4}=-22+\frac{81}{4}$$

Tāpiri -22 ki te \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=-\frac{7}{4}$$

Tauwehea x^{2}-9x+\frac{81}{4}. 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}.

$$\left(x-\frac{9}{2}\right)^{2}=-\frac{7}{4}$$

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

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{-\frac{7}{4}}$$

Whakarūnātia.

$$x-\frac{9}{2}=\frac{\sqrt{7}i}{2} x-\frac{9}{2}=-\frac{\sqrt{7}i}{2}$$

Me tāpiri \frac{9}{2} ki ngā taha e rua o te whārite.

$$x=\frac{9+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i+9}{2}$$