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

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

Whakaoti mō x

$$x=3$$

$$x=0$$

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

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

Whakamahia te āhuatanga tohatoha hei whakarea te x-1 ki te 2.

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

Pahekotia te 2x me x, ka 3x.

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

Tāpirihia te -2 ki te 1, ka -1.

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

Whakaarohia te \left(x-1\right)\left(x+1\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 1.

$$3x-1=x^{2}-1$$

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

$$3x-1-x^{2}=-1$$

Me tāpiri te 1 ki ngā taha e rua.

$$3x-1-x^{2}+1=0$$

Tāpirihia te -1 ki te 1, ka 0.

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

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.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 3 mō b, me 0 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 3^{2}.

$$x=\frac{-3±3}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-3±3}{-2}$$

Nā, me whakaoti te whārite x=\frac{-3±3}{-2} ina he tāpiri te ±. Tāpiri -3 ki te 3.

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

Whakawehe 0 ki te -2.

$$x=0$$

Nā, me whakaoti te whārite x=\frac{-3±3}{-2} ina he tango te ±. Tango 3 mai i -3.

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

Whakawehe -6 ki te -2.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=0 x=3$$

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

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

Whakamahia te āhuatanga tohatoha hei whakarea te x-1 ki te 2.

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

Pahekotia te 2x me x, ka 3x.

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

Tāpirihia te -2 ki te 1, ka -1.

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

Whakaarohia te \left(x-1\right)\left(x+1\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 1.

$$3x-1=x^{2}-1$$

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

$$3x-1-x^{2}=-1$$

Me tāpiri te 1 ki ngā taha e rua.

$$3x-x^{2}=-1+1$$

Tāpirihia te -1 ki te 1, ka 0.

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

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.

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

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+3x}{-1}=\frac{0}{-1}$$

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

$$x^{2}+\frac{3}{-1}x=\frac{0}{-1}$$

Whakawehe 3 ki te -1.

$$x^{2}-3x=\frac{0}{-1}$$

Whakawehe 0 ki te -1.

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

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=\frac{9}{4}$$

Tauwehea x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{9}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Whakarūnātia.

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

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=3 x=0$$