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

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

Whakaoti mō x

$$x=-1$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,2 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+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-4,x+2.

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

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

$$3+2x-4=\left(x-2\right)\left(x+2\right)$$

Tangohia te 4 i te 3, ka -1.

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

Whakaarohia te \left(x-2\right)\left(x+2\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 2.

$$-1+2x=x^{2}-4$$

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

$$-1+2x-x^{2}=-4$$

Me tāpiri te 4 ki ngā taha e rua.

$$-1+2x-x^{2}+4=0$$

Tāpirihia te -1 ki te 4, ka 3.

$$3+2x-x^{2}=0$$

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

$$-x^{2}+2x+3=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=-3=-3$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=3 b=-1$$

Tuhia anō te -x^{2}+2x+3 hei \left(-x^{2}+3x\right)+\left(-x+3\right).

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

Tauwehea te -x i te tuatahi me te -1 i te rōpū tuarua.

$$-x\left(x-3\right)-\left(x-3\right)$$

Whakatauwehea atu te kīanga pātahi x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-3\right)\left(-x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te -x-1=0.

$$x=3 x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,2 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+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-4,x+2.

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

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

$$3+2x-4=\left(x-2\right)\left(x+2\right)$$

Tangohia te 4 i te 3, ka -1.

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

Whakaarohia te \left(x-2\right)\left(x+2\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 2.

$$-1+2x=x^{2}-4$$

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

$$-1+2x-x^{2}=-4$$

Me tāpiri te 4 ki ngā taha e rua.

$$-1+2x-x^{2}+4=0$$

Tāpirihia te -1 ki te 4, ka 3.

$$3+2x-x^{2}=0$$

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.

$$-x^{2}+2x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 2 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-2±\sqrt{4+4\times 3}}{2\left(-1\right)}$$

Whakareatia 4 ki te 3.

$$x=\frac{-2±\sqrt{4+12}}{2\left(-1\right)}$$

Tāpiri 4 ki te 12.

$$x=\frac{-2±\sqrt{16}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 16.

$$x=\frac{-2±4}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-2±4}{-2}$$

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

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

Whakawehe 2 ki te -2.

$$x=-1$$

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

$$x=-\frac{6}{-2}$$

Whakawehe -6 ki te -2.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=-1 x=3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -2,2 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+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}-4,x+2.

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

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

$$3+2x-4=\left(x-2\right)\left(x+2\right)$$

Tangohia te 4 i te 3, ka -1.

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

Whakaarohia te \left(x-2\right)\left(x+2\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Pūrua 2.

$$-1+2x=x^{2}-4$$

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

$$-1+2x-x^{2}=-4$$

Me tāpiri te 1 ki ngā taha e rua.

$$2x-x^{2}=-4+1$$

Tāpirihia te -4 ki te 1, ka -3.

$$2x-x^{2}=-3$$

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.

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

Whakawehea ngā taha e rua ki te -1.

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

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

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

Whakawehe 2 ki te -1.

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

Whakawehe -3 ki te -1.

$$x^{2}-2x=3$$

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.

$$x^{2}-2x+1=3+1$$

Tāpiri 3 ki te 1.

$$x^{2}-2x+1=4$$

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}.

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

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

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

Whakarūnātia.

$$x-1=2 x-1=-2$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=3 x=-1$$