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

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

Whakaoti mō x

$$x = \frac{\sqrt{57} + 9}{2} \approx 8.274917218$$

$$x=\frac{9-\sqrt{57}}{2}\approx 0.725082782$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x.

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

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

$$x\times 4+2x-6=x\left(x-3\right)$$

Pahekotia te x\times 4 me 2x, ka 6x.

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

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

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

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

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

Me tāpiri te 3x ki ngā taha e rua.

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

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

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

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.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 9 mō b, me -6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-9±\sqrt{9^{2}-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Pūrua 9.

$$x=\frac{-9±\sqrt{81-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-9±\sqrt{81+4\left(-6\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -6.

$$x=\frac{-9±\sqrt{81-24}}{2\left(-1\right)}$$

Tāpiri 81 ki te -24.

$$x=\frac{-9±\sqrt{57}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-9±\sqrt{57}}{-2}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{57}}{-2} ina he tāpiri te ±. Tāpiri -9 ki te \sqrt{57}.

$$x=\frac{\sqrt{57}-9}{-2}$$

Whakawehe -9+\sqrt{57} ki te -2.

$$x=\frac{9-\sqrt{57}}{2}$$

Nā, me whakaoti te whārite x=\frac{-9±\sqrt{57}}{-2} ina he tango te ±. Tango \sqrt{57} mai i -9.

$$x=\frac{-\sqrt{57}-9}{-2}$$

Whakawehe -9-\sqrt{57} ki te -2.

$$x=\frac{\sqrt{57}+9}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{9-\sqrt{57}}{2} x=\frac{\sqrt{57}+9}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x-3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x.

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

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

$$x\times 4+2x-6=x\left(x-3\right)$$

Pahekotia te x\times 4 me 2x, ka 6x.

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

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

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

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

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

Me tāpiri te 3x ki ngā taha e rua.

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

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

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

Me tāpiri te 6 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

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

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.

$$-x^{2}+9x=6$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+9x}{-1}=\frac{6}{-1}$$

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

$$x^{2}+\frac{9}{-1}x=\frac{6}{-1}$$

Whakawehe 9 ki te -1.

$$x^{2}-9x=\frac{6}{-1}$$

Whakawehe 6 ki te -1.

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

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.

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

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

$$x^{2}-9x+\frac{81}{4}=-6+\frac{81}{4}$$

Tāpiri -6 ki te \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{57}{4}$$

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}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{57}{4}$$

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

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{57}{4}}$$

Whakarūnātia.

$$x-\frac{9}{2}=\frac{\sqrt{57}}{2} x-\frac{9}{2}=-\frac{\sqrt{57}}{2}$$

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

$$x=\frac{\sqrt{57}+9}{2} x=\frac{9-\sqrt{57}}{2}$$