Whakaoti i te 4/x-1+2/x+1=35

$$\frac { 4 } { x - 1 } + \frac { 2 } { x + 1 } = 35$$

Whakaoti mō x

$$x = \frac{2 \sqrt{326} + 3}{35} \approx 1.117455433$$

$$x=\frac{3-2\sqrt{326}}{35}\approx -0.946026862$$

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.

$$\left(x+1\right)\times 4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

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

$$4x+4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

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

$$4x+4+2x-2=35\left(x-1\right)\left(x+1\right)$$

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

$$6x+4-2=35\left(x-1\right)\left(x+1\right)$$

Tangohia te 2 i te 4, ka 2.

$$6x+2=35\left(x-1\right)\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 35 ki te x-1.

$$6x+2=\left(35x-35\right)\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 35x-35 ki te x+1 ka whakakotahi i ngā kupu rite.

$$6x+2=35x^{2}-35$$

Tangohia te 35x^{2} mai i ngā taha e rua.

$$6x+2-35x^{2}=-35$$

Me tāpiri te 35 ki ngā taha e rua.

$$6x+2-35x^{2}+35=0$$

Tāpirihia te 2 ki te 35, ka 37.

$$6x+37-35x^{2}=0$$

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.

$$-35x^{2}+6x+37=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -35 mō a, 6 mō b, me 37 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-35\right)\times 37}}{2\left(-35\right)}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-35\right)\times 37}}{2\left(-35\right)}$$

Whakareatia -4 ki te -35.

$$x=\frac{-6±\sqrt{36+140\times 37}}{2\left(-35\right)}$$

Whakareatia 140 ki te 37.

$$x=\frac{-6±\sqrt{36+5180}}{2\left(-35\right)}$$

Tāpiri 36 ki te 5180.

$$x=\frac{-6±\sqrt{5216}}{2\left(-35\right)}$$

Tuhia te pūtakerua o te 5216.

$$x=\frac{-6±4\sqrt{326}}{2\left(-35\right)}$$

Whakareatia 2 ki te -35.

$$x=\frac{-6±4\sqrt{326}}{-70}$$

Nā, me whakaoti te whārite x=\frac{-6±4\sqrt{326}}{-70} ina he tāpiri te ±. Tāpiri -6 ki te 4\sqrt{326}.

$$x=\frac{4\sqrt{326}-6}{-70}$$

Whakawehe -6+4\sqrt{326} ki te -70.

$$x=\frac{3-2\sqrt{326}}{35}$$

Nā, me whakaoti te whārite x=\frac{-6±4\sqrt{326}}{-70} ina he tango te ±. Tango 4\sqrt{326} mai i -6.

$$x=\frac{-4\sqrt{326}-6}{-70}$$

Whakawehe -6-4\sqrt{326} ki te -70.

$$x=\frac{2\sqrt{326}+3}{35}$$

Kua oti te whārite te whakatau.

$$x=\frac{3-2\sqrt{326}}{35} x=\frac{2\sqrt{326}+3}{35}$$

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.

$$\left(x+1\right)\times 4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

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

$$4x+4+\left(x-1\right)\times 2=35\left(x-1\right)\left(x+1\right)$$

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

$$4x+4+2x-2=35\left(x-1\right)\left(x+1\right)$$

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

$$6x+4-2=35\left(x-1\right)\left(x+1\right)$$

Tangohia te 2 i te 4, ka 2.

$$6x+2=35\left(x-1\right)\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 35 ki te x-1.

$$6x+2=\left(35x-35\right)\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 35x-35 ki te x+1 ka whakakotahi i ngā kupu rite.

$$6x+2=35x^{2}-35$$

Tangohia te 35x^{2} mai i ngā taha e rua.

$$6x+2-35x^{2}=-35$$

Tangohia te 2 mai i ngā taha e rua.

$$6x-35x^{2}=-35-2$$

Tangohia te 2 i te -35, ka -37.

$$6x-35x^{2}=-37$$

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.

$$-35x^{2}+6x=-37$$

Whakawehea ngā taha e rua ki te -35.

$$\frac{-35x^{2}+6x}{-35}=-\frac{37}{-35}$$

Mā te whakawehe ki te -35 ka wetekia te whakareanga ki te -35.

$$x^{2}+\frac{6}{-35}x=-\frac{37}{-35}$$

Whakawehe 6 ki te -35.

$$x^{2}-\frac{6}{35}x=-\frac{37}{-35}$$

Whakawehe -37 ki te -35.

$$x^{2}-\frac{6}{35}x=\frac{37}{35}$$

Whakawehea te -\frac{6}{35}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{35}. Nā, tāpiria te pūrua o te -\frac{3}{35} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{6}{35}x+\left(-\frac{3}{35}\right)^{2}=\frac{37}{35}+\left(-\frac{3}{35}\right)^{2}$$

Pūruatia -\frac{3}{35} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{6}{35}x+\frac{9}{1225}=\frac{37}{35}+\frac{9}{1225}$$

Tāpiri \frac{37}{35} ki te \frac{9}{1225} 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.

$$x^{2}-\frac{6}{35}x+\frac{9}{1225}=\frac{1304}{1225}$$

Tauwehea x^{2}-\frac{6}{35}x+\frac{9}{1225}. 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}.

$$\left(x-\frac{3}{35}\right)^{2}=\frac{1304}{1225}$$

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

$$\sqrt{\left(x-\frac{3}{35}\right)^{2}}=\sqrt{\frac{1304}{1225}}$$

Whakarūnātia.

$$x-\frac{3}{35}=\frac{2\sqrt{326}}{35} x-\frac{3}{35}=-\frac{2\sqrt{326}}{35}$$

Me tāpiri \frac{3}{35} ki ngā taha e rua o te whārite.

$$x=\frac{2\sqrt{326}+3}{35} x=\frac{3-2\sqrt{326}}{35}$$