Whakaoti i te 2x-3/x+1+2x-5/x-1=2

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

Whakaoti mō x

$$x=4$$

$$x=0$$

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.

$$\left(x-1\right)\left(2x-3\right)+\left(x+1\right)\left(2x-5\right)=2\left(x-1\right)\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-1 ki te 2x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}-5x+3+\left(x+1\right)\left(2x-5\right)=2\left(x-1\right)\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+1 ki te 2x-5 ka whakakotahi i ngā kupu rite.

$$2x^{2}-5x+3+2x^{2}-3x-5=2\left(x-1\right)\left(x+1\right)$$

Pahekotia te 2x^{2} me 2x^{2}, ka 4x^{2}.

$$4x^{2}-5x+3-3x-5=2\left(x-1\right)\left(x+1\right)$$

Pahekotia te -5x me -3x, ka -8x.

$$4x^{2}-8x+3-5=2\left(x-1\right)\left(x+1\right)$$

Tangohia te 5 i te 3, ka -2.

$$4x^{2}-8x-2=2\left(x-1\right)\left(x+1\right)$$

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

$$4x^{2}-8x-2=\left(2x-2\right)\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-2 ki te x+1 ka whakakotahi i ngā kupu rite.

$$4x^{2}-8x-2=2x^{2}-2$$

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

$$4x^{2}-8x-2-2x^{2}=-2$$

Pahekotia te 4x^{2} me -2x^{2}, ka 2x^{2}.

$$2x^{2}-8x-2=-2$$

Me tāpiri te 2 ki ngā taha e rua.

$$2x^{2}-8x-2+2=0$$

Tāpirihia te -2 ki te 2, ka 0.

$$2x^{2}-8x=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -8 mō b, me 0 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Tuhia te pūtakerua o te \left(-8\right)^{2}.

$$x=\frac{-\left(-8\right)±8}{2\times 2}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±8}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{8±8}{4}$$

Nā, me whakaoti te whārite x=\frac{8±8}{4} ina he tāpiri te ±. Tāpiri 8 ki te 8.

$$x=\frac{16}{4}$$

Whakawehe 16 ki te 4.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{8±8}{4} ina he tango te ±. Tango 8 mai i 8.

$$x=\frac{0}{4}$$

Whakawehe 0 ki te 4.

$$x=0$$

Kua oti te whārite te whakatau.

$$x=4 x=0$$

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.

$$\left(x-1\right)\left(2x-3\right)+\left(x+1\right)\left(2x-5\right)=2\left(x-1\right)\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x-1 ki te 2x-3 ka whakakotahi i ngā kupu rite.

$$2x^{2}-5x+3+\left(x+1\right)\left(2x-5\right)=2\left(x-1\right)\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te x+1 ki te 2x-5 ka whakakotahi i ngā kupu rite.

$$2x^{2}-5x+3+2x^{2}-3x-5=2\left(x-1\right)\left(x+1\right)$$

Pahekotia te 2x^{2} me 2x^{2}, ka 4x^{2}.

$$4x^{2}-5x+3-3x-5=2\left(x-1\right)\left(x+1\right)$$

Pahekotia te -5x me -3x, ka -8x.

$$4x^{2}-8x+3-5=2\left(x-1\right)\left(x+1\right)$$

Tangohia te 5 i te 3, ka -2.

$$4x^{2}-8x-2=2\left(x-1\right)\left(x+1\right)$$

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

$$4x^{2}-8x-2=\left(2x-2\right)\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x-2 ki te x+1 ka whakakotahi i ngā kupu rite.

$$4x^{2}-8x-2=2x^{2}-2$$

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

$$4x^{2}-8x-2-2x^{2}=-2$$

Pahekotia te 4x^{2} me -2x^{2}, ka 2x^{2}.

$$2x^{2}-8x-2=-2$$

Me tāpiri te 2 ki ngā taha e rua.

$$2x^{2}-8x=-2+2$$

Tāpirihia te -2 ki te 2, ka 0.

$$2x^{2}-8x=0$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-8x}{2}=\frac{0}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\left(-\frac{8}{2}\right)x=\frac{0}{2}$$

Whakawehe -8 ki te 2.

$$x^{2}-4x=\frac{0}{2}$$

Whakawehe 0 ki te 2.

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

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

$$x^{2}-4x+\left(-2\right)^{2}=\left(-2\right)^{2}$$

Pūrua -2.

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

Tauwehea x^{2}-4x+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}.

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

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

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

Whakarūnātia.

$$x-2=2 x-2=-2$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=4 x=0$$