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

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

Whakaoti mō x

$$x=-2$$

$$x=\frac{1}{2}=0.5$$

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.

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

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

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

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

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

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

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

Tangohia te 3 i te 3, ka 0.

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

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

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

Whakamahia te āhuatanga tuaritanga hei whakarea te -4x+4 ki te x+1 ka whakakotahi i ngā kupu rite.

$$6x=-4x^{2}+4$$

Me tāpiri te 4x^{2} ki ngā taha e rua.

$$6x+4x^{2}=4$$

Tangohia te 4 mai i ngā taha e rua.

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

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.

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

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

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

Pūrua 6.

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

Whakareatia -4 ki te 4.

$$x=\frac{-6±\sqrt{36-16\left(-4\right)}}{2\times 4}$$

Whakareatia -16 ki te -4.

$$x=\frac{-6±\sqrt{36+64}}{2\times 4}$$

Tāpiri 36 ki te 64.

$$x=\frac{-6±\sqrt{100}}{2\times 4}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-6±10}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-6±10}{8}$$

Nā, me whakaoti te whārite x=\frac{-6±10}{8} ina he tāpiri te ±. Tāpiri -6 ki te 10.

$$x=\frac{4}{8}$$

Whakahekea te hautanga \frac{4}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Nā, me whakaoti te whārite x=\frac{-6±10}{8} ina he tango te ±. Tango 10 mai i -6.

$$x=-\frac{16}{8}$$

Whakawehe -16 ki te 8.

$$x=-2$$

Kua oti te whārite te whakatau.

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

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.

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

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

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

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

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

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

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

Tangohia te 3 i te 3, ka 0.

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

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

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

Whakamahia te āhuatanga tuaritanga hei whakarea te -4x+4 ki te x+1 ka whakakotahi i ngā kupu rite.

$$6x=-4x^{2}+4$$

Me tāpiri te 4x^{2} ki ngā taha e rua.

$$6x+4x^{2}=4$$

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.

$$4x^{2}+6x=4$$

Whakawehea ngā taha e rua ki te 4.

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

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

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

Whakahekea te hautanga \frac{6}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Whakawehe 4 ki te 4.

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

Whakawehea te \frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{4}. Nā, tāpiria te pūrua o te \frac{3}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=1+\left(\frac{3}{4}\right)^{2}$$

Pūruatia \frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=1+\frac{9}{16}$$

Tāpiri 1 ki te \frac{9}{16}.

$$x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{25}{16}$$

Tauwehea x^{2}+\frac{3}{2}x+\frac{9}{16}. 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}.

$$\left(x+\frac{3}{4}\right)^{2}=\frac{25}{16}$$

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

$$\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{25}{16}}$$

Whakarūnātia.

$$x+\frac{3}{4}=\frac{5}{4} x+\frac{3}{4}=-\frac{5}{4}$$

Me tango \frac{3}{4} mai i ngā taha e rua o te whārite.

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