Whakaoti i te 2x/x-2=5+13x^2/x-2

$$\frac { 2 x } { x - 2 } = 5 + \frac { 13 x ^ { 2 } } { x - 2 }$$

Whakaoti mō x

$$x=-1$$

$$x=\frac{10}{13}\approx 0.769230769$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki 2 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-2.

$$2x=\left(x-2\right)\times 5+13x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 5.

$$2x=5x-10+13x^{2}$$

Tangohia te 5x mai i ngā taha e rua.

$$2x-5x=-10+13x^{2}$$

Pahekotia te 2x me -5x, ka -3x.

$$-3x=-10+13x^{2}$$

Tangohia te -10 mai i ngā taha e rua.

$$-3x-\left(-10\right)=13x^{2}$$

Ko te tauaro o -10 ko 10.

$$-3x+10=13x^{2}$$

Tangohia te 13x^{2} mai i ngā taha e rua.

$$-3x+10-13x^{2}=0$$

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

$$-13x^{2}-3x+10=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -13x^{2}+ax+bx+10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=-13\times 10=-130$$

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 -130.

$$1,-130 2,-65 5,-26 10,-13$$

Tātaihia te tapeke mō ia takirua.

$$1-130=-129 2-65=-63 5-26=-21 10-13=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=10 b=-13$$

Tuhia anō te -13x^{2}-3x+10 hei \left(-13x^{2}+10x\right)+\left(-13x+10\right).

$$\left(-13x^{2}+10x\right)+\left(-13x+10\right)$$

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

$$-x\left(13x-10\right)-\left(13x-10\right)$$

Whakatauwehea atu te kīanga pātahi 13x-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(13x-10\right)\left(-x-1\right)$$

Hei kimi otinga whārite, me whakaoti te 13x-10=0 me te -x-1=0.

$$x=\frac{10}{13} x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 2 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-2.

$$2x=\left(x-2\right)\times 5+13x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 5.

$$2x=5x-10+13x^{2}$$

Tangohia te 5x mai i ngā taha e rua.

$$2x-5x=-10+13x^{2}$$

Pahekotia te 2x me -5x, ka -3x.

$$-3x=-10+13x^{2}$$

Tangohia te -10 mai i ngā taha e rua.

$$-3x-\left(-10\right)=13x^{2}$$

Ko te tauaro o -10 ko 10.

$$-3x+10=13x^{2}$$

Tangohia te 13x^{2} mai i ngā taha e rua.

$$-3x+10-13x^{2}=0$$

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.

$$-13x^{2}-3x+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -13 mō a, -3 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-13\right)\times 10}}{2\left(-13\right)}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-13\right)\times 10}}{2\left(-13\right)}$$

Whakareatia -4 ki te -13.

$$x=\frac{-\left(-3\right)±\sqrt{9+52\times 10}}{2\left(-13\right)}$$

Whakareatia 52 ki te 10.

$$x=\frac{-\left(-3\right)±\sqrt{9+520}}{2\left(-13\right)}$$

Tāpiri 9 ki te 520.

$$x=\frac{-\left(-3\right)±\sqrt{529}}{2\left(-13\right)}$$

Tuhia te pūtakerua o te 529.

$$x=\frac{-\left(-3\right)±23}{2\left(-13\right)}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±23}{2\left(-13\right)}$$

Whakareatia 2 ki te -13.

$$x=\frac{3±23}{-26}$$

Nā, me whakaoti te whārite x=\frac{3±23}{-26} ina he tāpiri te ±. Tāpiri 3 ki te 23.

$$x=\frac{26}{-26}$$

Whakawehe 26 ki te -26.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{3±23}{-26} ina he tango te ±. Tango 23 mai i 3.

$$x=-\frac{20}{-26}$$

Whakahekea te hautanga \frac{-20}{-26} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{10}{13}$$

Kua oti te whārite te whakatau.

$$x=-1 x=\frac{10}{13}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 2 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x-2.

$$2x=\left(x-2\right)\times 5+13x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 5.

$$2x=5x-10+13x^{2}$$

Tangohia te 5x mai i ngā taha e rua.

$$2x-5x=-10+13x^{2}$$

Pahekotia te 2x me -5x, ka -3x.

$$-3x=-10+13x^{2}$$

Tangohia te 13x^{2} mai i ngā taha e rua.

$$-3x-13x^{2}=-10$$

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.

$$-13x^{2}-3x=-10$$

Whakawehea ngā taha e rua ki te -13.

$$\frac{-13x^{2}-3x}{-13}=-\frac{10}{-13}$$

Mā te whakawehe ki te -13 ka wetekia te whakareanga ki te -13.

$$x^{2}+\left(-\frac{3}{-13}\right)x=-\frac{10}{-13}$$

Whakawehe -3 ki te -13.

$$x^{2}+\frac{3}{13}x=-\frac{10}{-13}$$

Whakawehe -10 ki te -13.

$$x^{2}+\frac{3}{13}x=\frac{10}{13}$$

Whakawehea te \frac{3}{13}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{26}. Nā, tāpiria te pūrua o te \frac{3}{26} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{13}x+\left(\frac{3}{26}\right)^{2}=\frac{10}{13}+\left(\frac{3}{26}\right)^{2}$$

Pūruatia \frac{3}{26} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{3}{13}x+\frac{9}{676}=\frac{10}{13}+\frac{9}{676}$$

Tāpiri \frac{10}{13} ki te \frac{9}{676} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}+\frac{3}{13}x+\frac{9}{676}=\frac{529}{676}$$

Tauwehea x^{2}+\frac{3}{13}x+\frac{9}{676}. 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}.

$$\left(x+\frac{3}{26}\right)^{2}=\frac{529}{676}$$

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

$$\sqrt{\left(x+\frac{3}{26}\right)^{2}}=\sqrt{\frac{529}{676}}$$

Whakarūnātia.

$$x+\frac{3}{26}=\frac{23}{26} x+\frac{3}{26}=-\frac{23}{26}$$

Me tango \frac{3}{26} mai i ngā taha e rua o te whārite.

$$x=\frac{10}{13} x=-1$$