Whakaoti i te 1/x-3+18/x^2-9=x/x+3

$$\frac{ 1 }{ x-3 } + \frac{ 18 }{ { x }^{ 2 } -9 } = \frac{ x }{ x+3 }$$

Whakaoti mō x

$$x=7$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x^{2}-9,x+3.

$$x+3+18=\left(x-3\right)x$$

Tāpirihia te 3 ki te 18, ka 21.

$$x+21=\left(x-3\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-3 ki te x.

$$x+21=x^{2}-3x$$

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

$$x+21-x^{2}=-3x$$

Me tāpiri te 3x ki ngā taha e rua.

$$x+21-x^{2}+3x=0$$

Pahekotia te x me 3x, ka 4x.

$$4x+21-x^{2}=0$$

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

$$-x^{2}+4x+21=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=-21=-21$$

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. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -21.

$$-1,21 -3,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+21=20 -3+7=4$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=7 b=-3$$

Tuhia anō te -x^{2}+4x+21 hei \left(-x^{2}+7x\right)+\left(-3x+21\right).

$$\left(-x^{2}+7x\right)+\left(-3x+21\right)$$

Tauwehea te -x i te tuatahi me te -3 i te rōpū tuarua.

$$-x\left(x-7\right)-3\left(x-7\right)$$

Whakatauwehea atu te kīanga pātahi x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-7\right)\left(-x-3\right)$$

Hei kimi otinga whārite, me whakaoti te x-7=0 me te -x-3=0.

$$x=7 x=-3$$

Tē taea kia ōrite te tāupe x ki -3.

$$x=7$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x^{2}-9,x+3.

$$x+3+18=\left(x-3\right)x$$

Tāpirihia te 3 ki te 18, ka 21.

$$x+21=\left(x-3\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-3 ki te x.

$$x+21=x^{2}-3x$$

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

$$x+21-x^{2}=-3x$$

Me tāpiri te 3x ki ngā taha e rua.

$$x+21-x^{2}+3x=0$$

Pahekotia te x me 3x, ka 4x.

$$4x+21-x^{2}=0$$

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.

$$-x^{2}+4x+21=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 4 mō b, me 21 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\times 21}}{2\left(-1\right)}$$

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 21}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-4±\sqrt{16+4\times 21}}{2\left(-1\right)}$$

Whakareatia 4 ki te 21.

$$x=\frac{-4±\sqrt{16+84}}{2\left(-1\right)}$$

Tāpiri 16 ki te 84.

$$x=\frac{-4±\sqrt{100}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-4±10}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-4±10}{-2}$$

Nā, me whakaoti te whārite x=\frac{-4±10}{-2} ina he tāpiri te ±. Tāpiri -4 ki te 10.

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

Whakawehe 6 ki te -2.

$$x=-3$$

Nā, me whakaoti te whārite x=\frac{-4±10}{-2} ina he tango te ±. Tango 10 mai i -4.

$$x=-\frac{14}{-2}$$

Whakawehe -14 ki te -2.

$$x=7$$

Kua oti te whārite te whakatau.

$$x=-3 x=7$$

Tē taea kia ōrite te tāupe x ki -3.

$$x=7$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(x-3\right)\left(x+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o x-3,x^{2}-9,x+3.

$$x+3+18=\left(x-3\right)x$$

Tāpirihia te 3 ki te 18, ka 21.

$$x+21=\left(x-3\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-3 ki te x.

$$x+21=x^{2}-3x$$

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

$$x+21-x^{2}=-3x$$

Me tāpiri te 3x ki ngā taha e rua.

$$x+21-x^{2}+3x=0$$

Pahekotia te x me 3x, ka 4x.

$$4x+21-x^{2}=0$$

Tangohia te 21 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$4x-x^{2}=-21$$

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.

$$-x^{2}+4x=-21$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+4x}{-1}=-\frac{21}{-1}$$

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

$$x^{2}+\frac{4}{-1}x=-\frac{21}{-1}$$

Whakawehe 4 ki te -1.

$$x^{2}-4x=-\frac{21}{-1}$$

Whakawehe -21 ki te -1.

$$x^{2}-4x=21$$

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

$$x^{2}-4x+\left(-2\right)^{2}=21+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=21+4$$

Tāpiri 21 ki te 4.

$$x^{2}-4x+4=25$$

Tauwehea x^{2}-4x+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}.

$$\left(x-2\right)^{2}=25$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$x-2=5 x-2=-5$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=7 x=-3$$

Tē taea kia ōrite te tāupe x ki -3.

$$x=7$$