Whakaoti i te x/x+3=6/x-3-27-x^2/9-x^2

$$\frac { x } { x + 3 } = \frac { 6 } { x - 3 } - \frac { 27 - x ^ { 2 } } { 9 - x ^ { 2 } }$$

Whakaoti mō x

$$x = \frac{15}{2} = 7\frac{1}{2} = 7.5$$

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-3,9-x^{2}.

$$\left(x-3\right)x=\left(x+3\right)\times 6+27-x^{2}$$

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

$$x^{2}-3x=\left(x+3\right)\times 6+27-x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+3 ki te 6.

$$x^{2}-3x=6x+18+27-x^{2}$$

Tāpirihia te 18 ki te 27, ka 45.

$$x^{2}-3x=6x+45-x^{2}$$

Tangohia te 6x mai i ngā taha e rua.

$$x^{2}-3x-6x=45-x^{2}$$

Pahekotia te -3x me -6x, ka -9x.

$$x^{2}-9x=45-x^{2}$$

Tangohia te 45 mai i ngā taha e rua.

$$x^{2}-9x-45=-x^{2}$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$x^{2}-9x-45+x^{2}=0$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}-9x-45=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2x^{2}+ax+bx-45. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-9 ab=2\left(-45\right)=-90$$

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 -90.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -9.

$$a=-15 b=6$$

Tuhia anō te 2x^{2}-9x-45 hei \left(2x^{2}-15x\right)+\left(6x-45\right).

$$\left(2x^{2}-15x\right)+\left(6x-45\right)$$

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

$$x\left(2x-15\right)+3\left(2x-15\right)$$

Whakatauwehea atu te kīanga pātahi 2x-15 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-15\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-15=0 me te x+3=0.

$$x=\frac{15}{2} x=-3$$

Tē taea kia ōrite te tāupe x ki -3.

$$x=\frac{15}{2}$$

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-3,9-x^{2}.

$$\left(x-3\right)x=\left(x+3\right)\times 6+27-x^{2}$$

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

$$x^{2}-3x=\left(x+3\right)\times 6+27-x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+3 ki te 6.

$$x^{2}-3x=6x+18+27-x^{2}$$

Tāpirihia te 18 ki te 27, ka 45.

$$x^{2}-3x=6x+45-x^{2}$$

Tangohia te 6x mai i ngā taha e rua.

$$x^{2}-3x-6x=45-x^{2}$$

Pahekotia te -3x me -6x, ka -9x.

$$x^{2}-9x=45-x^{2}$$

Tangohia te 45 mai i ngā taha e rua.

$$x^{2}-9x-45=-x^{2}$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$x^{2}-9x-45+x^{2}=0$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}-9x-45=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -9 mō b, me -45 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 2\left(-45\right)}}{2\times 2}$$

Pūrua -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 2\left(-45\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-9\right)±\sqrt{81-8\left(-45\right)}}{2\times 2}$$

Whakareatia -8 ki te -45.

$$x=\frac{-\left(-9\right)±\sqrt{81+360}}{2\times 2}$$

Tāpiri 81 ki te 360.

$$x=\frac{-\left(-9\right)±\sqrt{441}}{2\times 2}$$

Tuhia te pūtakerua o te 441.

$$x=\frac{-\left(-9\right)±21}{2\times 2}$$

Ko te tauaro o -9 ko 9.

$$x=\frac{9±21}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{9±21}{4}$$

Nā, me whakaoti te whārite x=\frac{9±21}{4} ina he tāpiri te ±. Tāpiri 9 ki te 21.

$$x=\frac{30}{4}$$

Whakahekea te hautanga \frac{30}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{15}{2}$$

Nā, me whakaoti te whārite x=\frac{9±21}{4} ina he tango te ±. Tango 21 mai i 9.

$$x=-\frac{12}{4}$$

Whakawehe -12 ki te 4.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=\frac{15}{2} x=-3$$

Tē taea kia ōrite te tāupe x ki -3.

$$x=\frac{15}{2}$$

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-3,9-x^{2}.

$$\left(x-3\right)x=\left(x+3\right)\times 6+27-x^{2}$$

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

$$x^{2}-3x=\left(x+3\right)\times 6+27-x^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+3 ki te 6.

$$x^{2}-3x=6x+18+27-x^{2}$$

Tāpirihia te 18 ki te 27, ka 45.

$$x^{2}-3x=6x+45-x^{2}$$

Tangohia te 6x mai i ngā taha e rua.

$$x^{2}-3x-6x=45-x^{2}$$

Pahekotia te -3x me -6x, ka -9x.

$$x^{2}-9x=45-x^{2}$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$x^{2}-9x+x^{2}=45$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$2x^{2}-9x=45$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-9x}{2}=\frac{45}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}-\frac{9}{2}x=\frac{45}{2}$$

Whakawehea te -\frac{9}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{4}. Nā, tāpiria te pūrua o te -\frac{9}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{9}{2}x+\left(-\frac{9}{4}\right)^{2}=\frac{45}{2}+\left(-\frac{9}{4}\right)^{2}$$

Pūruatia -\frac{9}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{45}{2}+\frac{81}{16}$$

Tāpiri \frac{45}{2} ki te \frac{81}{16} 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.

$$x^{2}-\frac{9}{2}x+\frac{81}{16}=\frac{441}{16}$$

Tauwehea x^{2}-\frac{9}{2}x+\frac{81}{16}. 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}.

$$\left(x-\frac{9}{4}\right)^{2}=\frac{441}{16}$$

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

$$\sqrt{\left(x-\frac{9}{4}\right)^{2}}=\sqrt{\frac{441}{16}}$$

Whakarūnātia.

$$x-\frac{9}{4}=\frac{21}{4} x-\frac{9}{4}=-\frac{21}{4}$$

Me tāpiri \frac{9}{4} ki ngā taha e rua o te whārite.

$$x=\frac{15}{2} x=-3$$

Tē taea kia ōrite te tāupe x ki -3.

$$x=\frac{15}{2}$$