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

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

Whakaoti mō x

$$x=-4$$

$$x=2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+2.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+2 ki te 4.

$$4x+8-x\times 4=x\left(x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+2.

$$4x+8-x\times 4=x^{2}+2x$$

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

$$4x+8-x\times 4-x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$4x+8-x\times 4-x^{2}-2x=0$$

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

$$2x+8-x\times 4-x^{2}=0$$

Whakareatia te -1 ki te 4, ka -4.

$$2x+8-4x-x^{2}=0$$

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

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

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

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

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=-8=-8$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -8.

$$1,-8 2,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-8=-7 2-4=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=2 b=-4$$

Tuhia anō te -x^{2}-2x+8 hei \left(-x^{2}+2x\right)+\left(-4x+8\right).

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

Tauwehea te x i te tuatahi me te 4 i te rōpū tuarua.

$$x\left(-x+2\right)+4\left(-x+2\right)$$

Whakatauwehea atu te kīanga pātahi -x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+2\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te -x+2=0 me te x+4=0.

$$x=2 x=-4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+2.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+2 ki te 4.

$$4x+8-x\times 4=x\left(x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+2.

$$4x+8-x\times 4=x^{2}+2x$$

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

$$4x+8-x\times 4-x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$4x+8-x\times 4-x^{2}-2x=0$$

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

$$2x+8-x\times 4-x^{2}=0$$

Whakareatia te -1 ki te 4, ka -4.

$$2x+8-4x-x^{2}=0$$

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

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

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.

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

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

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-2\right)±\sqrt{4+4\times 8}}{2\left(-1\right)}$$

Whakareatia 4 ki te 8.

$$x=\frac{-\left(-2\right)±\sqrt{4+32}}{2\left(-1\right)}$$

Tāpiri 4 ki te 32.

$$x=\frac{-\left(-2\right)±\sqrt{36}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-\left(-2\right)±6}{2\left(-1\right)}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±6}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{2±6}{-2}$$

Nā, me whakaoti te whārite x=\frac{2±6}{-2} ina he tāpiri te ±. Tāpiri 2 ki te 6.

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

Whakawehe 8 ki te -2.

$$x=-4$$

Nā, me whakaoti te whārite x=\frac{2±6}{-2} ina he tango te ±. Tango 6 mai i 2.

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

Whakawehe -4 ki te -2.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=-4 x=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+2.

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

Whakamahia te āhuatanga tohatoha hei whakarea te x+2 ki te 4.

$$4x+8-x\times 4=x\left(x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+2.

$$4x+8-x\times 4=x^{2}+2x$$

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

$$4x+8-x\times 4-x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$4x+8-x\times 4-x^{2}-2x=0$$

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

$$2x+8-x\times 4-x^{2}=0$$

Tangohia te 8 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

Whakareatia te -1 ki te 4, ka -4.

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

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

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

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.

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

Whakawehea ngā taha e rua ki te -1.

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

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

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

Whakawehe -2 ki te -1.

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

Whakawehe -8 ki te -1.

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

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.

$$x^{2}+2x+1^{2}=8+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=8+1$$

Tāpiri 8 ki te 1.

$$x^{2}+2x+1=9$$

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}.

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

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x+1=3 x+1=-3$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=2 x=-4$$