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

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

Whakaoti mō x

$$x=\frac{2\sqrt{6}}{3}+1\approx 2.632993162$$

$$x=-\frac{2\sqrt{6}}{3}+1\approx -0.632993162$$

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.

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

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

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

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

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

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

$$6x+4-2=3\left(x-1\right)\left(x+1\right)$$

Tangohia te 2 i te 4, ka 2.

$$6x+2=3\left(x-1\right)\left(x+1\right)$$

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

$$6x+2=\left(3x-3\right)\left(x+1\right)$$

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

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

Tangohia te 3x^{2} mai i ngā taha e rua.

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

Me tāpiri te 3 ki ngā taha e rua.

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

Tāpirihia te 2 ki te 3, ka 5.

$$6x+5-3x^{2}=0$$

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.

$$-3x^{2}+6x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 6 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-3\right)\times 5}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-6±\sqrt{36+12\times 5}}{2\left(-3\right)}$$

Whakareatia 12 ki te 5.

$$x=\frac{-6±\sqrt{36+60}}{2\left(-3\right)}$$

Tāpiri 36 ki te 60.

$$x=\frac{-6±\sqrt{96}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te 96.

$$x=\frac{-6±4\sqrt{6}}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-6±4\sqrt{6}}{-6}$$

Nā, me whakaoti te whārite x=\frac{-6±4\sqrt{6}}{-6} ina he tāpiri te ±. Tāpiri -6 ki te 4\sqrt{6}.

$$x=\frac{4\sqrt{6}-6}{-6}$$

Whakawehe -6+4\sqrt{6} ki te -6.

$$x=-\frac{2\sqrt{6}}{3}+1$$

Nā, me whakaoti te whārite x=\frac{-6±4\sqrt{6}}{-6} ina he tango te ±. Tango 4\sqrt{6} mai i -6.

$$x=\frac{-4\sqrt{6}-6}{-6}$$

Whakawehe -6-4\sqrt{6} ki te -6.

$$x=\frac{2\sqrt{6}}{3}+1$$

Kua oti te whārite te whakatau.

$$x=-\frac{2\sqrt{6}}{3}+1 x=\frac{2\sqrt{6}}{3}+1$$

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.

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

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

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

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

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

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

$$6x+4-2=3\left(x-1\right)\left(x+1\right)$$

Tangohia te 2 i te 4, ka 2.

$$6x+2=3\left(x-1\right)\left(x+1\right)$$

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

$$6x+2=\left(3x-3\right)\left(x+1\right)$$

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

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

Tangohia te 3x^{2} mai i ngā taha e rua.

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

Tangohia te 2 mai i ngā taha e rua.

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

Tangohia te 2 i te -3, ka -5.

$$6x-3x^{2}=-5$$

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.

$$-3x^{2}+6x=-5$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}+6x}{-3}=-\frac{5}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\frac{6}{-3}x=-\frac{5}{-3}$$

Whakawehe 6 ki te -3.

$$x^{2}-2x=-\frac{5}{-3}$$

Whakawehe -5 ki te -3.

$$x^{2}-2x=\frac{5}{3}$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-2x+1=\frac{5}{3}+1$$

Tāpiri \frac{5}{3} ki te 1.

$$x^{2}-2x+1=\frac{8}{3}$$

Tauwehea x^{2}-2x+1. 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}.

$$\left(x-1\right)^{2}=\frac{8}{3}$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{8}{3}}$$

Whakarūnātia.

$$x-1=\frac{2\sqrt{6}}{3} x-1=-\frac{2\sqrt{6}}{3}$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{6}}{3}+1 x=-\frac{2\sqrt{6}}{3}+1$$