Whakaoti i te 2x-5/x^2+3x+2+4/x^2-4=1/x-2

$$\frac { 2 x - 5 } { x ^ { 2 } + 3 x + 2 } + \frac { 4 } { x ^ { 2 } - 4 } = \frac { 1 } { x - 2 }$$

Whakaoti mō x

$$x=6$$

Ngā Upane Whakamahi Tauwehe

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,-1,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-2\right)\left(x+1\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+3x+2,x^{2}-4,x-2.

$$\left(x-2\right)\left(2x-5\right)+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+4x+4=\left(x+1\right)\left(x+2\right)$$

Pahekotia te -9x me 4x, ka -5x.

$$2x^{2}-5x+10+4=\left(x+1\right)\left(x+2\right)$$

Tāpirihia te 10 ki te 4, ka 14.

$$2x^{2}-5x+14=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+14=x^{2}+3x+2$$

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

$$2x^{2}-5x+14-x^{2}=3x+2$$

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

$$x^{2}-5x+14=3x+2$$

Tangohia te 3x mai i ngā taha e rua.

$$x^{2}-5x+14-3x=2$$

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

$$x^{2}-8x+14=2$$

Tangohia te 2 mai i ngā taha e rua.

$$x^{2}-8x+14-2=0$$

Tangohia te 2 i te 14, ka 12.

$$x^{2}-8x+12=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-8x+12 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=12$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 12.

$$-1,-12 -2,-6 -3,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-6 b=-2$$

Me tuhi anō te kīanga whakatauwehe \left(x+a\right)\left(x+b\right) mā ngā uara i tātaihia.

$$\left(x-6\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x-2=0.

$$x=6 x=2$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,-1,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-2\right)\left(x+1\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+3x+2,x^{2}-4,x-2.

$$\left(x-2\right)\left(2x-5\right)+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+4x+4=\left(x+1\right)\left(x+2\right)$$

Pahekotia te -9x me 4x, ka -5x.

$$2x^{2}-5x+10+4=\left(x+1\right)\left(x+2\right)$$

Tāpirihia te 10 ki te 4, ka 14.

$$2x^{2}-5x+14=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+14=x^{2}+3x+2$$

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

$$2x^{2}-5x+14-x^{2}=3x+2$$

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

$$x^{2}-5x+14=3x+2$$

Tangohia te 3x mai i ngā taha e rua.

$$x^{2}-5x+14-3x=2$$

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

$$x^{2}-8x+14=2$$

Tangohia te 2 mai i ngā taha e rua.

$$x^{2}-8x+14-2=0$$

Tangohia te 2 i te 14, ka 12.

$$x^{2}-8x+12=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=1\times 12=12$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 12.

$$-1,-12 -2,-6 -3,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-6 b=-2$$

Tuhia anō te x^{2}-8x+12 hei \left(x^{2}-6x\right)+\left(-2x+12\right).

$$\left(x^{2}-6x\right)+\left(-2x+12\right)$$

Tauwehea te x i te tuatahi me te -2 i te rōpū tuarua.

$$x\left(x-6\right)-2\left(x-6\right)$$

Whakatauwehea atu te kīanga pātahi x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-6\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te x-2=0.

$$x=6 x=2$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,-1,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-2\right)\left(x+1\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+3x+2,x^{2}-4,x-2.

$$\left(x-2\right)\left(2x-5\right)+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+4x+4=\left(x+1\right)\left(x+2\right)$$

Pahekotia te -9x me 4x, ka -5x.

$$2x^{2}-5x+10+4=\left(x+1\right)\left(x+2\right)$$

Tāpirihia te 10 ki te 4, ka 14.

$$2x^{2}-5x+14=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+14=x^{2}+3x+2$$

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

$$2x^{2}-5x+14-x^{2}=3x+2$$

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

$$x^{2}-5x+14=3x+2$$

Tangohia te 3x mai i ngā taha e rua.

$$x^{2}-5x+14-3x=2$$

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

$$x^{2}-8x+14=2$$

Tangohia te 2 mai i ngā taha e rua.

$$x^{2}-8x+14-2=0$$

Tangohia te 2 i te 14, ka 12.

$$x^{2}-8x+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -8 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 12}}{2}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 12}}{2}$$

Whakareatia -4 ki te 12.

$$x=\frac{-\left(-8\right)±\sqrt{64-48}}{2}$$

Tāpiri 64 ki te -48.

$$x=\frac{-\left(-8\right)±\sqrt{16}}{2}$$

Tuhia te pūtakerua o te 16.

$$x=\frac{-\left(-8\right)±4}{2}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±4}{2}$$

Nā, me whakaoti te whārite x=\frac{8±4}{2} ina he tāpiri te ±. Tāpiri 8 ki te 4.

$$x=\frac{12}{2}$$

Whakawehe 12 ki te 2.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{8±4}{2} ina he tango te ±. Tango 4 mai i 8.

$$x=\frac{4}{2}$$

Whakawehe 4 ki te 2.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=6 x=2$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=6$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,-1,2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-2\right)\left(x+1\right)\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+3x+2,x^{2}-4,x-2.

$$\left(x-2\right)\left(2x-5\right)+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+\left(x+1\right)\times 4=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-9x+10+4x+4=\left(x+1\right)\left(x+2\right)$$

Pahekotia te -9x me 4x, ka -5x.

$$2x^{2}-5x+10+4=\left(x+1\right)\left(x+2\right)$$

Tāpirihia te 10 ki te 4, ka 14.

$$2x^{2}-5x+14=\left(x+1\right)\left(x+2\right)$$

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

$$2x^{2}-5x+14=x^{2}+3x+2$$

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

$$2x^{2}-5x+14-x^{2}=3x+2$$

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

$$x^{2}-5x+14=3x+2$$

Tangohia te 3x mai i ngā taha e rua.

$$x^{2}-5x+14-3x=2$$

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

$$x^{2}-8x+14=2$$

Tangohia te 14 mai i ngā taha e rua.

$$x^{2}-8x=2-14$$

Tangohia te 14 i te 2, ka -12.

$$x^{2}-8x=-12$$

Whakawehea te -8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -4. Nā, tāpiria te pūrua o te -4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-8x+\left(-4\right)^{2}=-12+\left(-4\right)^{2}$$

Pūrua -4.

$$x^{2}-8x+16=-12+16$$

Tāpiri -12 ki te 16.

$$x^{2}-8x+16=4$$

Tauwehea x^{2}-8x+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}.

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

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

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

Whakarūnātia.

$$x-4=2 x-4=-2$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$x=6 x=2$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=6$$