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

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

Whakaoti mō x

$$x=-1$$

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,x^{2}-4.

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

Tangohia te 4 i te 2, ka -2.

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

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.

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

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

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

Me tāpiri te 4 ki ngā taha e rua.

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

Tāpirihia te -2 ki te 4, ka 2.

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

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

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

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

$$a+b=1 ab=-2=-2$$

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.

$$a=2 b=-1$$

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

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

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

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

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

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

Hei kimi otinga whārite, me whakaoti te x-2=0 me te -x-1=0.

$$x=2 x=-1$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=-1$$

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,x^{2}-4.

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

Tangohia te 4 i te 2, ka -2.

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

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.

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

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

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

Me tāpiri te 4 ki ngā taha e rua.

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

Tāpirihia te -2 ki te 4, ka 2.

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

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.

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

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

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

Pūrua 1.

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

Whakareatia -4 ki te -1.

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

Whakareatia 4 ki te 2.

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

Tāpiri 1 ki te 8.

$$x=\frac{-1±\sqrt{9}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-1±3}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-1±3}{-2}$$

Nā, me whakaoti te whārite x=\frac{-1±3}{-2} ina he tāpiri te ±. Tāpiri -1 ki te 3.

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

Whakawehe 2 ki te -2.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{-1±3}{-2} ina he tango te ±. Tango 3 mai i -1.

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

Whakawehe -4 ki te -2.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=-1 x=2$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=-1$$

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,x^{2}-4.

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

Tangohia te 4 i te 2, ka -2.

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

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.

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

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

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

Me tāpiri te 2 ki ngā taha e rua.

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

Tāpirihia te -4 ki te 2, ka -2.

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

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.

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

Whakawehea ngā taha e rua ki te -1.

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

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

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

Whakawehe 1 ki te -1.

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

Whakawehe -2 ki te -1.

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

Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

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

Tāpiri 2 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{9}{4}$$

Tauwehea x^{2}-x+\frac{1}{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}.

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

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

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

Whakarūnātia.

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

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=2 x=-1$$

Tē taea kia ōrite te tāupe x ki 2.

$$x=-1$$